Appendix A Some Aspects of Semirings
Semirings considered as a common generalization of associative rings and dis- tributive lattices provide important tools in different branches of computer science. Hence structural results on semirings are interesting and are a basic concept. Semi- rings appear in different mathematical areas, such as ideals of a ring, as positive cones of partially ordered rings and fields, vector bundles, in the context of topolog- ical considerations and in the foundation of arithmetic etc. In this appendix some algebraic concepts are introduced in order to generalize the corresponding concepts of semirings N of non-negative integers and their algebraic theory is discussed.
A.1 Introductory Concepts
H.S. Vandiver gave the first formal definition of a semiring and developed the the- ory of a special class of semirings in 1934. A semiring S is defined as an algebra (S, +, ·) such that (S, +) and (S, ·) are semigroups connected by a(b+c) = ab+ac and (b+c)a = ba+ca for all a,b,c ∈ S.ThesetN of all non-negative integers with usual addition and multiplication of integers is an example of a semiring, called the semiring of non-negative integers. A semiring S may have an additive zero ◦ defined by ◦+a = a +◦=a for all a ∈ S or a multiplicative zero 0 defined by 0a = a0 = 0 for all a ∈ S. S may contain both ◦ and 0 but they may not coincide. Consider the semiring (N, +, ·), where N is the set of all non-negative integers;
a + b ={lcm of a and b, when a = 0,b= 0}; = 0, otherwise; and a · b = usual product of a and b. Then the integer 1 is the additive zero and integer 0 is the multiplicative zero of (N, +, ·).
M.R. Adhikari, A. Adhikari, Basic Modern Algebra with Applications, 585 DOI 10.1007/978-81-322-1599-8, © Springer India 2014 586 A Some Aspects of Semirings
In general, the concept of multiplicative zero 0 and additive zero ◦ may not coin- cide in a semiring. H.J. Weinert proved that both these concepts coincide if (S, +) is a cancellative semigroup. Clearly, S has an absorbing zero (or a zero element) iff it has elements 0 and ◦ which coincide. Thus an absorbing zero of a semiring S is an element 0, such that 0a = a0 = 0 and 0 + a = a + 0 = a for all a ∈ S. A semiring S may have an identity 1 defined by 1a = a1 = a, for all a ∈ S. A semiring S is said to be additively commutative, iff a + b = b + a for all a, b ∈ S. An additively commutative semiring with an absorbing zero 0 is called a hemiring. An additively cancellative hemiring is called a halfring. A multiplicatively cancellative commutative semiring with identity 1 is called a semidomain. We mainly consider semirings S for which (S, +) is commutative. If (S, ·) is also commutative, S is called a commutative semiring. Moreover, to avoid trivial exceptions, each semiring S is assumed to have at least two elements. A subset A = ∅ of a semiring S is called an ideal (left, right) of S iff a + b ∈ A,sa ∈ A and as ∈ A(sa ∈ A,as ∈ A) hold, for all a,b ∈ A and all s ∈ S. An ideal A of S is called proper, iff A ⊂ S holds, where ⊂ denotes proper inclusion, and a proper ideal A is called maximal, iff there is no ideal B of S satisfying A ⊂ B ⊂ S. An ideal A of S is called trivial iff A = S holds or A ={0} (the latter clearly holds if S has an absorbing zero 0). Many results in rings related to ideals have no analogues in semirings. As noted by Henriksen, it is not the case that any ideal A of a semiring S is the kernel of a homomorphism. To get rid of these difficulties, he defined in 1958, a more restricted class of ideals in a semiring which he called k-ideals. A k-ideal A of a semiring S is an ideal of S such that whenever x + a ∈ A, where a ∈ A and x ∈ S, then x ∈ A. Iizuka defined a still more restricted class of ideals in semirings, which he called h-ideals. An h-ideal A of a semiring S is an ideal of S such that if x + a + u = b + u, where x,u ∈ S and a,b ∈ A, then x ∈ A. It is clear that every h-ideal is a k- ideal. But examples disapprove the converse. Using only commutativity of addition, the following concepts and statements essentially due to Bourne, Zassenhaus, and Henriksen, are well known. For each ideal A of a semiring S,thek-closure A¯ = ¯ ¯ ¯ {a ∈ S : a + a1 = a2 for some ai ∈ A} is a k-ideal of S satisfying A ⊆ A and A = A. Clearly, an ideal A of S is a k-ideal of S iff A = A¯ holds. A proper k-ideal A of S is called a maximal k-ideal of S iffthereisnok-ideal B of S satisfying A ⊂ B ⊂ S. An ideal A of a semiring S is called completely prime or prime iff ab ∈ A implies a ∈ A or b ∈ A, for all a,b ∈ S. Let S and T be hemirings. A map f : S → T is said to be a hemiring homomor- phism iff f(s1 + s2) = f(s1) + f(s2); f(s1s2) = f(s2)f (s2) and f(0) = f(0). A hemiring homomorphism f : S → T is said to be an N-homomorphism, iff whenever f(x)= f(y)for some x,y ∈ S, there exist ri ∈ ker f such that x + r1 = y + r2 holds. An equivalence relation ρ on a semiring S is called a congruence relation iff (a, b) ∈ ρ implies (a + c,b + c) ∈ ρ,(ca,cb) ∈ ρ and (ac,bc) ∈ ρ for all c ∈ S. A congruence relation ρ on a semiring S is said to be additively cancellative (AC) iff (a + c, b + d) ∈ ρ, and (c, d) ∈ ρ imply (a, b) ∈ ρ. Each ideal A of S defines a congruence relation ρA on (S, +, ·) given by ρA ={(x, y) ∈ S × S : x + a1 = A.1 Introductory Concepts 587 y + a2 for some ai ∈ A}; this is known as Bourne congruence. The corre- sponding class semiring S/ρA, consisting of the classes xρA, is also denoted by S/A. Each ideal A of S defines another type of congruence relation known as Iizuka congruence defined by σA ={(x, y) ∈ S × S : x + a + u = y + b+u for some a,b ∈ A and u ∈ S}. The corresponding class semiring S/σA consists of the classes xσA,x∈ S. Bourne and Zassenhaus defined the zeroid Z(S) of a semiring S as Z(S) ={z ∈ S : z + x = x for some x ∈ S};itisanh-ideal of S and is contained in every h-ideal of S. Z(S) and S are called trivial h-ideals and other h-ideals (if they exist) are called proper h-ideals of S. A proper h-ideal A of S is called a maximal h-ideal iff there is no h-ideal B of S satisfying A ⊂ B ⊂ S.Theh-closure A¯h of each ideal ¯h A of a semiring S defined by A ={x ∈ S : x + a1 + u = a2 + u for some ai ∈ A and u ∈ S} is the smallest h-ideal of S containing A. A semiring S is said to be additively regular iff for each a ∈ S, there exists an element b ∈ S such that a = a + b + a. If in addition, the element b is unique and satisfies b = b + a + b, then S is called an additively inverse semiring. A semiring S is said to be semisubtractive iff for each pair a, b in S at least one of the equations a + x = b or b + x = a is solvable in S. Let S be a semiring with absorbing zero 0. A left semimodule over S is a com- mutative additive semigroup M with a zero element 0 together with an operation
S × M → M; (a, x) → ax called the scalar multiplication such that
for all a,b ∈ S,x,y ∈ M, a(x + y) = ax + ay, (a + b)x = ax + bx, (ab)x = a(bx), 0x = 0.
A right S-semimodule is defined in an analogous manner. Let H and T be hemir- ings. An (H, T ) bi-semimodule M is both a left semimodule over H and a right semimodule over T such that (bx)a = b(xa) for all x ∈ M, b ∈ H and a ∈ T . Let R be an arbitrary hemiring. Let S be also a hemiring. R is said to be an S-semialgebra iff R is a bi-semimodule over S such that (ax)b = a(xb), for all a,b ∈ R and x ∈ S. An h-ideal A of the S-semialgebra R is said to be a left modular h-ideal iff there exists an element e ∈ R such that (i) each x ∈ R satisfies ex + a + z = x + b + z for some z ∈ R and some a,b ∈ A; (ii) each s ∈ S satisfies se+ c + h = es + d + h for some h ∈ R and some c,d ∈ A. In this case e is called a left unit modulo A. A pair (e1,e2) of elements of a semiring S is called an identity pair iff a + e1a = e2a and a + ae1 = ae2, for all a ∈ S. We define a right modular h-ideal in a similar manner. A congruence ρ on a semiring S is called a ring congruence iff the quotient semiring S/ρ is a ring. A semiring S who ◦ and 1 is said to be h-Noetherian (k-Noetherian) iff it satis- fies any one of the following three equivalent conditions: 588 A Some Aspects of Semirings
(i) S satisfies the ascending chain condition on h-ideals (k-ideals); (ii) The maximal condition for h-ideals (k-ideals) holds in S; (iii) Every h-ideal (k-ideal) in S is finitely generated.
A.2 More Results on Semirings
In this section we present more results of algebraic theory of semirings which are closely related to the corresponding results of ring theory.
Theorem A.2.1 Let S be a semiring such that S = a1,a2,...,an is finitely gen- erated. Then each proper k-ideal A of S is contained in a maximal k-ideal of S.
Proof Let k(A) be the set of all k-ideals B of S satisfying A ⊆ B S, partially ordered by inclusion. Then k(A) = ∅,forAitself belongs to k(A). Consider a chain { ∈ } = Bi,i I in k(A). We claim that B i∈I Bi is a proper k-ideal of S.Let a,b ∈ B and s ∈ S. Then there exist i, j ∈ I such that a ∈ Bi and b ∈ Bj .As {Bi,i ∈ I} forms a chain, either Bi ⊆ Bj or Bj ⊆ Bi . For definiteness, we sup- pose Bi ⊆ Bj , so that both a,b ∈ Bj . Since Bj is a k-ideal of S, a + b ∈ Bj ⊆ B, as, sa ∈ Bj ⊆ B for all s ∈ S.Again,ifa + x ∈ B, a ∈ B, x ∈ S, then proceeding as above, a + x,a ∈ Bj for some Bj of the above chain. Since Bj is a k-ideal, it follows that x ∈ Bj ⊆ B.AsaresultB is a k-ideal of S. Next we verify that B is a proper k-ideal of S. Suppose to the contrary B = S = a1,a2,...,an . Then each { } = ar would belong to some k-ideal Bir of the chain Bi , where r 1, 2,...,n. There ∈ being only finitely many Bir ’s, one contains all others, let us call it Bt , while t I . Thus a1,a2,...,an all lie in this Bt . Consequently, Bt = S, which is clearly impos- sible. As a result B ∈ k(A). Thus by Zorn’s Lemma k(A) has a maximal element as we were to prove.
Corollary Let S be a semiring with 1. Then each proper k-ideal of S is contained inamaximalk-ideal of S.
Proof The proof is immediate by S = 1 .
We now consider conditions on a semiring S such that S has non-trivial k-ideals or maximal k-ideals, among others by the help of the congruence class semiring S/A defined by an ideal A of S.LetS denote S = S\{0},ifS has 0, and S = S, otherwise.
Definition A.2.1 A semiring S is said to satisfy the condition (C ) iff for all a ∈ S and all s ∈ S, there are s1,s2 ∈ S such that s + s1a = s2a holds.
Clearly, if S has an identity 1, then (C ) is equivalent to the condition (C), which states that 1 + s1a = s2a holds, for each a ∈ S and suitable s1,s2 ∈ S. A.2 More Results on Semirings 589
Example A.2.1 Let Q+ be the set of all non-negative rational numbers. Then (Q+, +, ·) with usual operations is a semiring with 1 as identity satisfying con- dition (C). The same is true, more generally, for each positive cone P of a totally ordered skew fields (Fuchs 1963).
Example A.2.2 Let N be the set of all non-negative integers. Define a + b = max{a,b} and denote by ab the usual multiplication. Then (N, +, ·) is a semiring with 1 as identity which satisfies (C), since 1 + a = a holds for all a ∈ N+.
Lemma A.2.1 If a semiring S with an absorbing zero 0 satisfies condition (C ), then ab = 0 for a,b ∈ S implies a = 0 or b = 0.
Proof By way of contradiction, assume ab = 0 and a = 0 = b. Then s + s1a = s2a according to (C ) yields sb + s1ab = s2ab, i.e., sb = 0 for all s ∈ S. Consequently x + s3b = s4b implies x = 0, for all s3,s4 ∈ S, which contradicts (C ) applied to the element b ∈ S .
Theorem A.2.2 Let S be a semiring. Then condition (C ) implies that S contains only trivial k-ideals. The converse is true if (S, ·) is commutative and provided that S has an element 0, then Sa ={sa : s ∈ S}={0} holds for all a ∈ S .
Proof Assume that S satisfies (C ).LetA be a k-ideal of S which contains at least one element a ∈ S . Then s + s1a = s2a according to (C ) implies s ∈ A, for each s ∈ S, i.e., A = S. For the converse, our supplementary assumption on S shows that Sa is an ideal of S and that Sa = {0} holds for each a ∈ S if S has an element 0. We assume that S has only trivial k-ideals. Then the k-ideal Sa coincides with S for each a ∈ S , regardless whether S has an element 0 or not. Now Sa ={s ∈ S : s + s1a = s2a for some si ∈ S}=S states that S satisfies condition (C ).
Theorem A.2.3 Let S be a commutative semiring with identity 1 and A a proper k-ideal of S. Then A is maximal iff the semiring S/A = S/ρA satisfies condition (C), where ρA is the Bourne congruence on S.
Proof Suppose A is a maximal k-ideal of S. Then A is the absorbing zero of S/A and 1ρA its identity. Consider any cρA ∈ (S/A) . Then c ∈ A holds and the small- est ideal B of S containing c and A consists of all elements sc, a and sc + a for s ∈ S and a ∈ A.FromA B it follows B = S and hence 1 + b1 = b2 for suitable elements b1,b2 ∈ B.Ifb1 = s1c + a1 and b2 = s2c + a2 for suit- able si ∈ S and ai ∈ A, then we obtain 1 + s1c + a1 = s2c + a2. This implies 1ρA + (s1ρA)(cρA) = (s2ρA)(cρA). Discussions of other cases are similar. As a result S/ρA satisfies condition (C). Conversely, assume (C) for S/A and let B be a k-ideal of S satisfying A B.
Then there is an element c ∈ B/A, and cρA ∈ (S\A) yields 1ρA + (s1ρA)(cρA) = (s2ρA)(cρA) for suitable si ∈ S by (C). Hence (1 + s1c)ρA = (s2c)ρA. Conse- quently, 1 + s1c + a1 = s2c + a2 holds for some ai ∈ A. Hence 1 + b1 = b2 holds 590 A Some Aspects of Semirings for some b1,b2 ∈ B. This implies 1 ∈ B. Consequently, B = S. This shows that A is a maximal k-ideal of S.
[For more results, see Weinert et al. 1996]. In the rest of this section we only consider semirings S in which addition is commutative.
Definition A.2.2 An ideal A of a semirings S is said to be completely prime iff ab ∈ A implies a ∈ A or b ∈ B, for any a,b ∈ S.
Proposition A.2.1 Let S be a commutative semiring with identity. Then each maxi- mal k-ideal A of S is completely prime.
Proof By Theorem A.2.3, the semiring S/A satisfies the conditions (C) and hence condition (C ) (see Definition A.2.1). Since S/A has A as its absorbing zero, we can apply Lemma A.2.1 and find that S/A has no zero divisors. Hence aρA = A and bρA = A imply (ab)ρA = A, i.e., a ∈ A and b ∈ A imply ab ∈ A, where ρA is the Bourne Congruence defined in Sect. A.1.
Concerning the converse of Proposition B.2.1, we show that a completely prime ideal A of a commutative semiring S with identity needs not be a k-ideal, and if it is one, A needs not be a maximal k-ideal of S.
Example A.2.3 (a) Let S be the set of all real numbers a satisfying 0 number r such that 0
All maximal k-ideals of the semiring N are described below:
Proposition A.2.2 Let (N, +, ·) be the semiring of non-negative integers under usual operations. Then N has exactly the k-ideals a ={na : n ∈ N} for each a ∈ N. Consequently, the maximal k-ideals of N are given by p for each positive prime p. A.3 Ring Congruences and Their Characterization 591
Proof Clearly, each ideal a of N is a k-ideal. Now assume that A = {0} is a k- ideal of N.Leta be the smallest positive integer contained in A, and b any element of A. Then b = qa + r holds for some positive q and r ∈ N satisfying 0 ≤ r Remark 1 None of the maximal k-ideals p of N is a maximal ideal of N.This follows since each ideal A = p is properly contained in the proper ideal B ={b ∈ N : b ≥ p} of N. Remark 2 Let A = p , p is prime in Remark 1 and τA be the hull-kernel topology defined in A. Then (a) (A, τA) is a connected space, (b) (A, τA) is compact. (See Proposition A.6.1.) Recall that a semiring S is said to be additively regular iff for each a ∈ S, ∃ an element b ∈ S such that a = a + b + a. If in addition, the element b is unique and satisfies b = b + a + b, then S is called an additively inverse semiring. In an additively inverse semiring, the unique inverse b of an element a is usually denoted by a . Exercise (Karvellas 1974) Let S be an additively inverse semiring. Then (i) x = (x ) , (x + y) = y + x , (xy) = x y = xy and xy = x y ∀x,y ∈ S. (ii) E+ ={x ∈ S : x + x = x} is an additively commutative semilattice and an ideal of S. Definition A.2.3 Let S be an additively inverse semiring in which addition is com- mutative and E+ denote the set of all additive idempotents of S.Aleftk-ideal A of S is said to be full iff E+ ⊆ A. A right k-ideal of S is defined dually. A non-empty subset I of S is called a full k-ideal iff I is both a left and a right full k-ideal. Example A.2.4 (a) In a ring every ring ideal is a full k-ideal. (b) In a distributive lattice with more than two elements, a proper ideal is a k-ideal but not a full k-ideal. (c) Z × N+ ={(a, b) : a,b are integers and b>0}. Define (a, b) + (c, d) = (a + c,lcm(b, d)) and (a, b)(c, d) = (ac, gcd(b, d)). Then Z × N+ becomes an additively inverse semiring in which addition is commutative. Let A ={(a, b) ∈ Z × N+ : a = 0}. Then A is a full k-ideal of Z × N+. A.3 Ring Congruences and Their Characterization We now define a ring congruence and characterize those ring congruences ρ on additively inverse semirings S in which addition is commutative and is such that 592 A Some Aspects of Semirings −(aρ) = a ρ, where a denotes the inverse of a in S and −(aρ) denotes the additive inverse of aρ in the ring S/ρ. Definition A.3.1 A congruence ρ on a semiring S is called a ring congruence iff the quotient semiring S/ρ is a ring. Theorem A.3.1 Let A be a full k-ideal of S. Then the relation ρA ={(a, b) ∈ S ×S : a + b ∈ A} is a ring congruence of S such that −(aρA) = a ρA. + Proof Since a + a ∈ E ⊆ A for all a ∈ S, it follows that ρA is reflexive. Let a + b ∈ A. Now from W.O. Ex. 1(a), we find that (a + b ) ∈ A. Then b + a = (b ) + a = (a + b ) ∈ A. Hence ρA is symmetric. Let a + b ∈ A and b + c ∈ A. Then a + b + b + c ∈ A.Alsob + b ∈ E+ ⊆ A. Since A is a k-ideal, we find that a + c ∈ A. Hence ρA is an equivalence relation. Let (a, b) ∈ ρA and c ∈ S. Then a + b ∈ A. Since (c + a) + (c + b) = c + a + b + c = a + b + c + c ∈ A, ca + (cb) = ca + cb = c a + b ∈ A and ac + (bc) = ac + b c = a + b c ∈ A, it follows that ρA is a congruence on S. So we obtain the quotient semiring where addition and multiplication are defined by aρA + bρA = (a + b)ρA and (aρA)(bρA) = (ab)ρA. Now aρA + bρA = (a + b)ρA = (b + a)ρA = bρA + aρA. Let e ∈ E+ and a ∈ S.Now(e + a) + a = e + (a + a ) ∈ E+. We find that (e + a)ρA = aρA. Then eρA + aρA = aρA. Also aρA + a ρA = a + a ρA = eρA. Hence eρA is the zero element and a ρA is the negative element of aρA in the ring S/ρA. Theorem A.3.2 Let ρ be a congruence of S such that S/ρ is a ring and −(aρ) = a ρ. Then there exists a full k-ideal A of S such that ρA = ρ. Proof Let A ={a ∈ S : (a, e) ∈ ρ for some e ∈ E+}. Since ρ is reflexive, it follows that E+ ⊆ A. Then A = ∅, since E+ = ∅.Leta,b ∈ A. Then there exist e,f ∈ E+ such that (a, e) ∈ ρ and (b, f ) ∈ ρ. Then (a +b,e+f)∈ ρ.Bute +f ∈ E+. Hence a + b ∈ A.Againforanyr ∈ S, (ra, re) ∈ ρ and (ar, er) ∈ ρ.Butre and er ∈ E+. Hence A is an ideal of S.Leta + b ∈ A and b ∈ A. Then there exist e,f ∈ E+ such that (a + b,f ) ∈ ρ and (b, e) ∈ ρ. Hence fρ = (a + b)ρ = aρ + bρ = aρ + eρ . A.4 k-Regular Semirings 593 But fρ and eρ are additive idempotents in the ring S/ρ. Hence eρ = fρ is the zero element of S/ρ.Asaresultaρ is the zero element of S/ρ. Then aρ = eρ .This implies a ∈ A. So we find that A is a full k-ideal of S. Consider now the congruences + ρA and ρ.Let(a, b) ∈ ρ. Then (a +b ,b+b ) ∈ ρ.Butb +b ∈ E . Hence a +b ∈ A and (a, b) ∈ ρA. Conversely, suppose that (a, b) ∈ ρA. Then a + b ∈ A. Hence (a + b ,e)∈ ρ for some e ∈ E+. As a result, eρ = aρ + b ρ = aρ − bρ holds in the ring S/ρ.Buteρ is the zero element of S/ρ. Consequently aρ = bρ . This show that (a, b) ∈ ρ and hence ρA = ρ. Remark (a) Certain types of ring congruences on an additive inverse semiring are characterized with the help of full k-ideals (see Theorems A.3.1 and A.3.2). (b) It is shown that the set of all full k-ideals of an additively inverse semiring in which addition is commutative forms a complete lattice which is also modular (see W.O. Ex. 1(d)). A.4 k-Regular Semirings Definition A.4.1 Let S be an additively commutative semiring and a ∈ S. Then a is called regular (in the sense of Von Neuman) if a = axa holds for some x ∈ S and k-regular if there are x,y ∈ S satisfying a + aya = axa. If all the elements of S have the corresponding property, S is called a regular or a k-regular semiring. Since aya can be added to both sides of a = axa, each regular semiring is also k-regular. If S happens to be a ring, both concepts coincide and each (additively commutative) semifield S is regular. Example A.4.1 For an additively commutative semiring S with an identity 1, a con- dition C is introduced in Definition A.2.1 by the property that each element a ∈ S which is not multiplicatively absorbing zero satisfies 1 + s1a = s2a for suitable ele- ments si ∈ S. This condition implies that S has only trivial k-ideals, and the converse holds if (S, ·) is commutative (see Theorem A.2.2). Clearly, each semiring of this kind is k-regular, but not necessarily regular. Lemma A.4.1 Let S be an additively commutative and cancellative semiring and R = D(S) its difference ring. If R is regular, then S is k-regular. The converse holds if S is semisubtractive, but not in general. Proof Let R be regular and a ∈ S. Then a = ara holds for a suitable element r ∈ R, say r = x − y for x,y ∈ S, which yields a + aya = axa. Now assume that S is k-regular and semisubtractive. Then each r = 0ofR satisfies r = a or r =−a for some a ∈ S, and a + aya = axa for some x,y ∈ S implies a = a(x − y)a or −a = a(y − x)a in R. For the last statement, let Q[z] be the polynomial ring over the field of rational numbers and R = Q[z]/(z2). For brevity, we may write a + bz for all elements of R. It was shown in Weinert (1963) that S ={a + bz ∈ R : a>0} 594 A Some Aspects of Semirings is a subsemifield of R such that R = D(S) holds. Hence S is even regular, whereas zrz = 0 for all r ∈ R shows that R is not regular. Note that S can be replaced by S ∪{0} in this example. Example A.4.2 Let Mn,n(H0) be the semiring of all n × n matrices over the semi- field H0 of all non-negative rational numbers. Then Mn,n(H0) is k-regular, since the matrix ring Mn,n(Q) = D(Mn,n(H0)) is well known to be regular. Clearly, H0 may be replaced in this example by any semifield S such that D(S) is a field [see Weinert (1963)]. Lemma A.4.2 Let S be a k-regular semiring. Then A ∩ B = BA holds for each left k-ideal A and each right k-ideal B of S. Proof From BA ⊆ A ∩ B, we get BA ⊆ A¯ and BA ⊆ B = B. Hence BA ⊆ A ∩ B. To show the reverse inclusion, assume a ∈ A ∩ B. Then aza = a(za) ∈ BA for each z ∈ S and a + aya = axa imply a ∈ BA. Theorem A.4.1 For such a hemiring S, the set K(S) of all k-ideals of S forms a complete lattice (K(S), ⊆). This lattice is distributive if (2.1) A∩B = AB holds for all k-ideals A = A¯ and B = B of S, hence in particular if the hemiring is k-regular. Proof The intersection of any set of k-ideals of S is known to be a k-ideal of S, and S is a k-ideal of S. This yields the result that (K(S) ⊆) is a complete lattice, for which A ∧ B = A ∩ B is the infimum and A ∨ B = A + B is the supremum of A,B ∈ K(S). Moreover, as in each lattice, A ∧ (B ∨ C) ⊇ (A ∧ B) ∨ (A ∧ C) holds for all A,B,C ∈ K(S). So it remains to show the reverse inclusion. Now A ∧ (B ∨ C) = A ∩ (B + C) = A(B + C) holds by (2.1). Suppose x ∈ A(B + C). Then x + u = v for some u, v ∈ A(B + C). Hence u = a1y1 and v = a2v2 for suitable ai ∈ A and yi ∈ B + C. Then y1 + b1 + c1 = b2 + c2 and y2 + b3 + c3 = b4 + c4 for suitable bi ∈ B and ci ∈ C. Thus a1y1 + a1b1 + a1c1 = a1b2 + a1c2 yields u + a1b1 + a1c1 = a1b2 + a1c2, which states u ∈ AB + AC. Likewise we obtain v ∈ AB + AC and thus x ∈ AB + AC for x + u = v, since AB + AC is k-closed. Now A ∧ (B ∨ C) ⊆ AB + AC and AB + AC ⊆ AB + AC = (A ∩ B)+ (A ∩ C) = (A ∧ B)∨ (A ∧ C) complete the proof that the lattice (K(S), ⊆) is distributive. We show in this context that, for each hemiring S, the condition (2.1) is equiva- lent to (2.2) A = AA holds for each k-ideal A = A¯ of S. A.4 k-Regular Semirings 595 Clearly, (2.1) for A = B yields (2.2), and AB ⊆ A ∩ B implies for k-ideals al- ways AB ⊆ A ∩ B = A ∩ B. Applying (2.2) to the k-ideal A ∩ B, we get the other inclusion of (2.1) by A ∩ B = (A ∩ B)(A ∩ B) ⊆ AB. Remark (2.2) follows from the (obviously stronger) condition that A = A2 holds for all ideals of a hemiring S. The latter, for instance is satisfied if S is regular, implied by a result of Ashan (1993) that the complete lattice of all ideals of S is Brouwerian and thus distributive. However, none of the sufficient conditions for distributivity mentioned so far is necessary. Example A.4.3 One easily checks that the set S ={0,a,b} becomes a hemiring according to the following tables. + 0 ab • 0 ab 0 0 ab 0 000 a aab a 000 b bbb b 00b This hemiring has the k-ideals {0}⊂{0,a}⊂S and one more ideal {0,b} whose k-closure is S. Hence (K(S), ⊆) and obviously also the lattice of all ideals of S are distributive. However, {0,a}2 ={0} disproves (2.2) and thus k-regularity and clearly A2 = A for all ideals A of S. Finally considering any ring as a hemiring, the k-ideals of R are just the ring- theoretical ideals of R, well known to form a modular lattice. So one could suspect that the latter holds also for the lattice K(S) for each hemiring S. But the following example shows that the complete lattice (K(S), ⊆) of all k-ideals of a hemiring S does not need to be modular. Remark For each k-regular hemiring S,thesetK(S) of all k-ideals of S forms a complete distributive lattice. In general, however, the lattice of all k-ideals of a hemiring does not need to be even modular. Example A.4.4 Consider the set S ={0,a,b,s,c,d}. We define (S, +) by the table + 0 abscd 0 0 abscd a aassdd b bsbsdd s ssssdd c cdddcd d dddddd and xy = 0 for all x,y ∈ S. Instead of dealing with associativity of (S, +) directly, note that (S\{0}, +) is the commutative and idempotent semigroup generated by {a,b,c,d} subject to relations a + c = d and b + c = d. Clearly, s is shorting for a + b. Now one checks that the subsets of S as shown in Fig. A.1 are k-ideals of this 596 A Some Aspects of Semirings Fig. A.1 k-ideals of S S {0,a,b,s} {0,a} {0,c} {0} Fig. A.2 Commutativity of ◦ the ring completion diagram S f g h S R semiring. By a well known criterion, {0,a,b,s}∩{0,c}={0} and {0,a}∨{0,c}= {0,a}+{0,c}={0,a,c,d}=S show that (K(S) ⊆) is not modular. Remark For one sided h-ideals and k-ideals in semirings, see Weinert et al. (1996). A.5 The Ring Completion of a Semiring The concept of a ring completion of a semiring arises through the process of passing from the semiring of non-negative integers to yield the ring of integers. This concept is used to study the stability properties of vector bundles (see Husemoller 1966, Chap. 8). Definition A.5.1 Let S be a semiring with 0 (absorbing). The ring completion of S is a pair (S◦,f), where S◦ is a ring and f : S → S◦ is a semiring homomorphism such that for any semiring homomorphism h : S → R, where R is an arbitrary ring, there exists a unique ring homomorphism g : S◦ → R such that g ◦ f = h, i.e., the diagram in Fig. A.2 is commutative. We now prescribe a construction of S◦. Construction of S◦: Consider the pairs (a, b) ∈ S × S and define an equivalence relation ρ on these pairs: (a, b) and (c, d) are said to be ρ-equivalent iff there exists A.6 Structure Spaces of Semirings 597 Fig. A.3 Commutative ◦ diagram for ring completion S f S g h f1 ◦ S1 some l ∈ S such that a +d +l = c +b +l. Denote the ρ-equivalent class of (a, b) by a,b . We may consider a,b =a − b.LetS◦ be the set of all ρ-equivalence class a,b . Define a,b + c,d = a +c,b+d and a,b · c,d = ac+bd,bc+ad . These compositions are independent of the choice of the representatives of classes and hence are well defined. Clearly, S◦, +, · is a ring with its zero element 0 = 0, 0 and negative of a,b being b,a . The homomorphism f : S → S◦ is defined by f(x)= x,0 . ◦ ◦ ◦ Uniqueness of (S ,f):Let (S1 ,f1) be another ring completion of (S ,f). Then : ◦ → ◦ : ◦ → ◦ there exist ring homomorphisms g S S1 and h S1 S making the diagram in Fig. A.3 commutative. It follows from the commutativity of the diagram that both g ◦ h and h ◦ g are ◦ ◦ identity homomorphisms of rings. Hence the rings S and S1 are isomorphic. Remark 1 S◦ may be viewed as the free abelian group generated by the set S mod- ulo the subgroup generated by (a + b) + (−1)a + (−1)b, a,b ∈ S. Remark 2 The process of passing from a semigroup to a group is analogous and yields its group completion. A.6 Structure Spaces of Semirings The structure spaces of semirings, formed by the class of prime k-ideals and prime full k-ideals are considered in this section. More precisely, the properties such as separation axioms, compactness and connectedness in these structure spaces are studied. Finally, these properties for the semiring of non-negative integers are ex- amined. In this section we only consider semirings S for which (S, +) is commutative. Definition A.6.1 Let A = ∅ be any subset of a semiring S. Then the ideal generated by A is the intersection of all ideals I of S containing A. We recall the definition of hull-kernel topology and some of its properties (Gill- man 1957; Kohis 1957; Slowikowski and Zawadowski 1955). 598 A Some Aspects of Semirings Definition A.6.2 Let A be the set of all proper prime ideals of a commutative ring A ¯ ¯ ={ ∈ A : ⊆ } with identity. For any subset A of , A given by A I I ∈A Iα I gives ¯ α A → A a closure operator defining some topology τA called the hull-kernel topol- ogy on A. The hull-kernel topology on the set A of all proper prime k-ideals (full k-ideals) of a commutative semiring with identity is defined in a similar way. Proposition A.6.1 (a) (i) A ⊆ A¯, ¯ (ii) A¯ = A¯, (iii) A ⊆ B ⇒ A¯ ⊆ B¯ , and (iv) A ∪ B = A¯ ∪ B¯ for all subsets A, B of A (b) (i) For any commutative semiring S with identity 1, let A be the set of all proper prime k-ideals of S and for each a ∈ S, Δ(a) ={I ∈ A : a ∈ I}. If cΔ(a) = A \ Δ(a) then {cΔ(a) : a ∈ S} forms an open base for the hull- kernel topology on A. (ii) (A,τA ) is a T1-space iff no element of A is contained in any other element of A. (c) Let M be the set of all maximal k-ideals of S. Then (M,τM ) is a T1-space, where τM is the induced topology on M from (A,τA ). (d) For the semiring N of non-negative integers, let A = p , p is prime and A be the set of all prime ideals A of N. Then (i) (A,τA ) is a connected space; (ii) (A,τA ) is compact; where τA is the hull-kernel topology on A. Proof Left as an exercise. We now study further properties of the structure spaces. Let I be any k-ideal of S. Define Δ(I) ={I ∈ A : I ⊆ I }. Proposition A.6.2 AnyclosedsetinA is of the form Δ(I), where I is a k-ideal of S. ¯ Proof Let A be any closed set in A, where A ⊆ A.LetA ={Iα : α ∈ Λ} and I = ¯ = α∈Λ Iα. Then A Δ(I). Theorem A.6.1 (A,τA ) is a Hausdorff space if and only if for any distinct pair of elements I , J of A, there exist a,b ∈ S such that a ∈ J , b ∈ I and there does not exist any element K of A such that a ∈ K and b ∈ K. Proof Let (A,τA ) be Hausdorff. Then for any pair of distinct elements I , J of A there exist basic open sets cΔ(a) and cΔ(b) such that I ∈ cΔ(a), J ∈ cΔ(b) and cΔ(a) ∩ cΔ(b) =∅. It follows that a ∈ I , b ∈ J and for any prime k-ideal K of S A.6 Structure Spaces of Semirings 599 for which a,b ∈ K implies K ∈ cΔ(a) ∩ cΔ(b), a contradiction, since cΔ(a) ∩ cΔ(b) =∅. Conversely, let the given condition hold and I,J ∈ A, I = J .Leta,b ∈ S be such that a ∈ I,b ∈ J and there does not exist any K of A such that a ∈ K, b ∈ K. Then I ∈ cΔ(a), J ∈ cΔ(b) and cΔ(a) ∩ cΔ(b) =∅, which proves that (A,τA ) is Hausdorff. Corollary If (A,τA ) is a T2-space, then no proper prime k-ideal contains any other proper prime k-ideal. If (A,τA ) contains more than one element, then there exist a,b ∈ S such that A = cΔ(a) ∪ cΔ(b) ∪ Δ(I), where I is the k-ideal generated by a, b. Proof Suppose (A,τA ) is a T2-space. Since every T2-space is a T1-space, (A,τA ) is a T1-space. Hence by Proposition A.6.1 no proper prime k-ideal contains any other proper prime k-ideal. Now let I,J ∈ A, where I = J . Then by Theorem A.6.1, there exist a, b in S such that a = b,a ∈ I,b ∈ J and cΔ(a) ∩ cΔ(b) =∅.LetI be the k-ideal generated by a, b. Then since cΔ(a) ∩ Δ(b) =∅. Any element of A, belongs to cΔ(a) or cΔ(b) or Δ(I), and therefore cΔ(a) ∪ cΔ(b) ∪ Δ(I) = A. Theorem A.6.2 (A,τA ) is a regular space if and only if for any I ∈ A and a ∈ I , a ∈ S, there exist a k-ideal J of S and b ∈ S such that I ∈ cΔ(b) ⊆ Δ(J ) ⊆ cΔ(a). Proof Let (A,τA ) be a regular space. Then for any I ∈ A and any closed set Δ(J ) not containing I , there exist disjoint open sets U,V such that I ∈ U and Δ(J ) ⊆ V . If a ∈ I , then I ∈ cΔ(a) and A \ cΔ(a) is a closed set not containing I . Hence U,V are disjoint open sets containing I and A \ cΔ(a), respectively. Then there exists b ∈ S such that I ∈ cΔ(b) ⊆ Δ(J ) ⊆ cΔ(a), where J is a k-ideal of S. Conversely, let the given condition hold. Let I ∈ A and Δ(K) be any closed set not containing I .Leta ∈ I,a ∈ K. Then by the given condition, there exist a k-ideal J and an element b ∈ S such that I ∈ cΔ(b) ⊆ Δ(J ) ⊆ cΔ(a). Obviously, cΔ(a) ∩ Δ(K) =∅. So, cΔ(b) and A \ Δ(J ) are two disjoint open sets containing I and Δ(K), respectively. Consequently, (A,τA ) is a regular space. Theorem A.6.3 (A,τA ) is a compact space if and only if for any collection {ai} of elements of S there exists a finite number of elements a1,a2,...,ar in S such that for any I ∈ A, there exists ai such that ai ∈ I . Proof Let (A,τA ) be a compact space. Then the open cover {cΔ(a) : a ∈ S} has a finite subcover {cΔ(ai) : i = 1,...,r}. Thus if I ∈ A, then I ∈ cΔ(ai) for some i which implies that ai ∈ I . Hence ai,...,ar are the required finite number of ele- ments. The converse follows from Kohis (1957, Lemma 3.1). Corollary If S is finitely generated, then (A,τA ) is compact. 600 A Some Aspects of Semirings Proof Let {ai : i = 1,...,r} be a set of generators of S. Then for any I ∈ A, there exists ai such that ai ∈ I , since I is a proper k-ideal. Hence by Theorem A.6.3. (A,τA ) is compact. Proposition A.6.3 Let (S,τS ) be the space of all proper prime full k-ideals of S. + Then (S,τS ) is compact if and only if E ={x ∈ S : x + x = x}={0}. Proof Let {Δ(Iα) : α ∈ Λ} be any collection of closed sets in S with finite inter- section property. Let I be the proper prime k-ideal which is also the full k-ideal + + generated by E . Since any prime full k-ideal J contains E ,J contains I . Hence ∈ = ∅ S I α∈Λ Δ(Iα) . Consequently, ( ,τS ) is compact. Definition A.6.3 A semiring S is said to be k-Noetherian if and only if it satisfies the ascending chain condition on k-ideals on S i.e., if and only if for the ascending chain I1 ⊆ I2 ⊆···⊆In ··· of k-ideals of S, there exists a positive integer m such that In = Im for all n ≥ m. Theorem A.6.4 If S is a k-Noetherian semiring, then (A,τA ) is countably com- pact. { }∞ A Proof Let Δ(In) n=1 be a countable collection of closed sets in , with finite in- tersection property. Let A denote the prime k-ideal generated by A, where A(=∅) is any subset of A. Let us consider the following ascending chain of prime k-ideals: I1 ⊆ I1 ∪ I2 ⊆ I1 ∪ I2 ∪ I3 ⊆···. Since S is k-Noetherian, there exists a positive ∪ ∪···∪ = ∪ ∪···∪ =··· integer m such that I1 I2 In I1 I2 Im+1 , which shows ∪···∪ ∈ ∞ = ∅ A that I1 Im n=1 Δ(In) . Hence ( ,τA ) is countably compact. Corollary If S is a k-Noetherian semiring and (A,τA ) is second countable, then (A,τA ) is compact. Proof It follows from Theorem A.6.4 and the fact any open cover of a second count- able space has a countable subcover. Theorem A.6.5 (A,τA ) is disconnected if and only if there exist a k-ideal I of S and a collection of points {aα}α∈Λ of S not belonging to I such that if I ∈ A and aα ∈ I , ∀α ∈ Λ, then I \ I = ∅. Proof Let (A,τA ) be not connected. Then there exists a non-trivial open and closed subset ofA.LetI be the k-ideal of S for which Δ(I) is closed as well as open. Then = { } Δ(I) α∈Λ cΔ(aα), where aα α∈λ is a collection of points of S. Now since cΔ(aα) ⊆ Δ(I), ∀α ∈ Λ for any Iα ∈ cΔ(aα) we have I ⊆ Iα, and therefore aα ∈ I as aα ∈ Iα, ∀α ∈ Λ.NowforanyI ∈ A and aα ∈ I , ∀α ∈ Λ we have I ∈ Δ(I). Consequently, I ⊆ I i.e., I \ I = ∅. = Conversely, let the given condition hold. Then Δ(I) α∈Λ cΔ(aα) is an open and closed non-trivial subset of A and hence (A,τA ) is disconnected. A.7 Worked-Out Exercises and Exercises 601 + Proposition A.6.4 (S,τS ) is connected space if E = {0}. Proof Let I be the proper k-ideal generated by E+. Then I belongs to any closed set Δ(I ) of A, since any full k-ideal of S contains E+. Consequently any two closed sets of A are not disjoint. Hence (S,τS ) is connected. Example A.6.1 Let S = N be the set semiring of non-negative integers with respect to usual addition and multiplication. Then the prime proper k-ideals of S are (p) = {np : n ∈ N} where p is a prime number (Sen and Adhikari 1993). Let A ={(p) : p is prime} and τA be the hull-kernel topology defined in A. Since any prime proper k-ideal of N is not contained in other prime proper k-ideal, (A,τA ) is a T1-space. Now let (p1) and (p2) be two distinct elements of A and let n1,n2 be two elements of S such that n1 ∈ (p1) and n2 ∈ (p2), then we can always find an element (p) of A, such that n1 ∈ (p) and n2 ∈ (p). In fact one can take p>n1,n2,p1,p2 and p is prime. Hence by Theorem A.6.1, (A,τA ) is not a Hausdorff space. cΔ(n) is infinite and its complement, which is closed in A, is finite. So any two non-trivial open sets intersect. Consequently, (A,τA ) is neither a T2-space nor a regular space. For this reason (A,τA ) is a connected space. Clearly, (A,τA ) is compact by Theorem A.6.3. A.7 Worked-Out Exercises and Exercises Worked-Out Exercises (W.O. Ex) 1. Let S be an additively inverse semiring in which addition is commutative and E+ denote the set of all additive idempotents of S. (a) Every k-ideal of S is an additively inverse subsemiring of S. (b) Let A be an ideal of S. Then (i) A¯ ={a ∈ S : a + x ∈ A for some x ∈ A} is a k-ideal of S such that A ⊆ A¯. (ii) A¯ = A ⇔ A is a k-ideal. (c) Let A and B be full k-ideals of S, then A + B is a full k-ideal of S such that A ⊆ A + B and B ⊆ A + B. (d) If I(S) denotes the set of full k-ideals of S, then I(S) is a complete lattice which is also modular. Solution (a) Let I be a k-ideal of S. Clearly I is a subsemiring of S.Leta ∈ I . Then a + a + a = a ∈ I. Since I is a k-ideal, if follows a + a ∈ I . Again this implies that a ∈ I . Hence (a) follows. (b) Let a,b ∈ A¯. Then a + x,b+ y ∈ A for some x,y ∈ A.Nowa + x + b + y = (a + b) + (x + y) ∈ A. 602 A Some Aspects of Semirings As x + y ∈ A, a + b ∈ A¯.Nextletr ∈ S, ra + rx = r(a + x) ∈ A. As rx ∈ A, ra ∈ A¯. Similarly, ar ∈ A¯.AsaresultA¯ is an ideal of S. Next, let c and c + d ∈ A¯. Then there exist x and y in A such that c + x ∈ A and c + d + y ∈ A. Now d + (c + x + y) = (c + d + y) + x ∈ A and c + x + y ∈ A. Hence d ∈ A¯ and A¯ is a k-ideal of S. Since a + a ∈ A for all a ∈ A, it follows that A ⊆ A¯. (c) It can be shown that A + B is an ideal of S. Then from (b), we find A + B is a k-ideal and A + B ⊆ A + B.NowE+ ⊆ A,B. Hence E+ ⊆ A + B ⊆ A + B. This implies that A + B is a full k-ideal. Let a ∈ A. Then + a = a + a + a = a + a + a ∈ A + B as a + a ∈ E ⊆ B. Hence A ⊆ A + B and similarly B ⊆ A + B. (d) We first note that I(S) is a partially ordered set with respect to usual set inclusion. Let A,B ∈ I(S). Then A ∩ B ∈ I(S)and from (c), A + B ∈ I(S). Define A ∧ B = A ∩ B and A ∨ B = A + B.LetC ∈ I(S) be such that A,B ⊆ C. Then A + B ⊆ C and A + B ⊆ C.ButC = C. Hence A + B ⊆ C.AsaresultA + B is the lub of A, B. Thus we find that I(S) is a lattice. Now E+ is an ideal of S. Hence E+ ∈ I(S)and also S ∈ I(S); consequently I(S)is a complete lattice. Next suppose that A,B,C ∈ I(S)such that A ∧ B = A ∧ C and A ∨ B = A ∨ C and B ⊆ C. Let x ∈ C. Then x ∈ A ∨ C = A ∨ B = A + B. Hence there exists a + b ∈ A + B such that x + a + b = a1 + b1 for some a1 ∈ A, b1 ∈ B. Then x + a + a + b = a1 + b1 + a . Now x ∈ C,a + a ∈ C and b ∈ B ⊆ C. Hence a1 + b1 + a ∈ C.Butb1 ∈ C. Consequently, a1 + a ∈ C ∩ A = C ∩ B. Hence a1 + a ∈ B.Sofromx + a + b = a1 + b1 we find that x + a + a + b = a1 + a + b ∈ B.But(a + a ) + b ∈ B is a k-ideal. Hence x ∈ B and B = C. This proves that I(S)is a modular lattice. Exercises 1. Define homomorphism, monomorphism, epimorphism, isomorphism for semir- ings with the help of the corresponding definitions for semigroups and rings. 2. A mapping f : (S, +, ·) → (T , +, ·) is a semiring homomorphism iff both f : (S, +) → (T , +) and f : (S, ·) → (T , ·) are semigroup homomorphisms. 3. Let f : (S, +, ·) → (T , +, ·) be a semiring homomorphism. Then the homo- morphic image (f (S), +, ·) of (S, +, ·) is also a semiring. If (S, +, ·) is com- mutative, then (f (S), +, ·) is also commutative. If f is an isomorphism then f −1 is also so. 4. Let H be a hemiring and 0 = a ∈ H and (e1,e2) = (0, 0) be an identity pair of H . A pair (c, d) ∈ H × H is called an inverse relative of the identity pair (e1,e2) iff e1 + ca = e2 + da and e1 + ac = e2 + ad. An additively cancellative hemiring H is called a division hemiring iff H contains an identity pair and every non-zero element of H is invertible in H . A hemiring H is said to be a left k-Artinian (left Artinian) iff H satisfies the d.c.c for k-ideals (ideals) of H . A.8 Additional Reading 603 (a) Let H = ab : a,b,d ∈ R and a,d ≥ 5 ∪ 00 is a division hemiring 0d 00 50 , 60 with 05 06 as a left identity pair. (b) A division hemiring H has no zero divisors. (c) A division hemiring is multiplicatively cancellative. (d) Let H be an additively cancellative hemiring with more than one element. Then H is multiplicatively cancellative k-Artinian iff H is a division hemir- ing. 5. A semiring (S, +, ·) satisfying |S|≥2 is called a semifield iff the set of its non-zero elements S forms a multiplicative group. (a) Let (S, +, ·) be a semifield satisfying |S |≥2. Then the identity e of the group (S , ·) is also the identity of (S, +, ·) and (S, +, ·) is multiplicatively cancellative. (b) Let (S, +, ·) be a semiring satisfying |S |≥2. Then (S, +, ·) is a semifield iff it satisfies one of the following statements: (i) (S, +, ·) has a left identity e and each a ∈ S is invertible in (S, ·); (ii) For arbitrary elements a ∈ S ,b ∈ S, ∃ elements x,y ∈ S such that ax = b and ya = b. (c) Let (S, +, ·) be a semiring satisfying |S|≥2. If (S, +, ·) has an identity and each a ∈ S has a (left) inverse a ∈ S, then (S, +, ·) is semifield. 6. A semiring S satisfying the condition (C ) of Definition A.2.1 contains at most one left ideal consisting of a single element. 7. Let S = (S, +) be a semimodule and A ⊂ S a maximal k-closed subsemimodule of S. Then A is also k-closed in S i.e., A ⊂ B = B ⊆ S ⇒ B = S for each k- closed subsemimodule B of S. 8. If a semiring S satisfies the condition (C ) of Definition A.2.1, it contains at most two left k-ideals, which are in fact two-sided ones, namely S and possibly, the ideal {0} consisting of the multiplicatively absorbing element 0 of S. 9. Let S be a semiring. The each proper k-ideals (h-ideal) of S is contained in a maximal k-ideal (h-ideal) of S if ∃ a finitely generated ideal I of S such that I¯ = S. 10. Give an example of multiplicatively commutative semiring S which has only two h-ideals (viz. Z(S) and S), but an infinite chain of k-ideal Bi satisfying A = Z(S) ⊂ B1 ⊂ B2 ⊂···⊂S. (In particular, A = Z(S) is a maximal h-ideal of S, but not a maximal k-ideal.) A.8 Additional Reading We refer the reader to the books (Adhikari and Adhikari 2003; Adhikari and Das 1994; Adhikari et al. 1996; Ahsan 1993; Fuchs 1963; Gillman 1957; Glazek 604 A Some Aspects of Semirings 1985; Golan 1992; Hazewinkel 1996; Hebich and Weinert 1993; Henriksen 1958; Husemoller 1966; Iizuka 1959; Sen and Adhikari 1992, 1993; Vandiver 1934; Wein- ert 1963; Weinert et al. 1996) for further details. References Adhikari, M.R., Adhikari, A.: Groups, Rings and Modules with Applications, 2nd edn. Universities Press, Hyderabad (2003) Adhikari, M.R., Das, M.K.: Structure spaces of semirings. Bull. Calcutta Math. Soc. 86, 313Ð317 (1994) Adhikari, M.R., Sen, M.K., Weinert, H.J.: On k-regular semirings. Bull. Calcutta Math. Soc. 88, 141Ð144 (1996) Ahsan, J.: Fully idempotent semirings. Proc. Jpn. Acad., Ser. A 69, 185Ð188 (1993) Fuchs, L.: Partially Ordered Algebraic Systems. Addison-Wesley, Reading (1963) Gillman, L.: Rings with Hausdorff structure space. Fundam. Math. 45, 1Ð16 (1957) Glazek, K.: A Short Guide Through the Literature on Semirings. Math. Inst. Univ. Wroclaw, Poland (1985) Golan, J.S.: The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Longman, Harlow (1992) Hazewinkel, M. (ed.): Handbook of Algebra, vol. I. Elsevier, Amsterdam (1996) Hebich, U., Weinert, H.J.: Algebraische Theorie und Anwendungen in der Informatik. Tember, Stuttgart (1993) Henriksen, M.: Ideals in semirings with commutative addition. Not. Am. Math. Soc. 5, 321 (1958) Husemoller, D.: Fibre Bundles, 2nd edn. Springer, New York (1966) Iizuka, K.: On the Jacobson radical of a semiring. Tohoku Math. J. 11(2), 409Ð421 (1959) Karvellas, P.H.: Inverse semirings. J. Aust. Math. Soc. 18, 277Ð288 (1974) Kohis, C.W.: The space of pair ideals of a ring. Fundam. Math. 45, 1Ð27 (1957) Sen, M.K., Adhikari, M.R.: On k-ideals of semirings. Int. J. Math. Math. Sci. 15(2), 347Ð350 (1992) Sen, M.K., Adhikari, M.R.: On maximal k-ideals of semirings. Proc. Am. Math. Soc. 113(3), 699Ð 703 (1993) Slowikowski, W., Zawadowski, W.: A generalization of maximal ideals method of Stone and Gelfaand. Fundam. Math. 42, 215Ð231 (1955) Vandiver, H.S.: Note on a simply type of algebra in which cancellation law of addition does not hold. Bull. Am. Math. Soc. 40, 914Ð920 (1934) Weinert, H.J.: Über Halbringe und Halbkorper II. Acta Math. Acad. Sci. Hung. 14, 209Ð227 (1963) Weinert, H.J., Sen, M.K., Adhikari, M.R.: One sided k-ideals and h-ideals in semirings. Pannon. Hungary 7(1), 147Ð162 (1996) Appendix B Category Theory Category theory is a very important branch of modern mathematics. It has been growing quite rapidly both in contents and applicability to other branches of the sub- ject. The concepts of categories, functors, natural transformations and duality form the foundation of category theory. These concepts were introduced during 1942Ð 1945 by S. Eilenberg and S. Mac Lane.1 Originally, the purpose of these notions was to provide a technique for classifying certain concepts such as that of natural isomorphism. In pedagogical methods in mathematics we compartmentalize math- ematics into its different branches without emphasizing their interrelationships. But with the help of category theory one can move from one branch of mathematics to another. It provides a convenient language to tie together several notions and exist- ing results of different branches of mathematics in a unified way. This language is conveyed in this chapter through modern algebra, algebraic topology, topological algebra, sheaf theory etc. B.1 Categories A category may be thought roughly as consisting of sets, possibly with additional structures, and functions, possibly preserving additional structures. More precisely, a category can be defined by the following characteristics. Definition B.1.1 A category C consists of (a) a class of objects X,Y,Z,... denoted by ob(C); (b) for each ordered pair of objects X, Y a set of morphisms with domain X and range Y denoted by C(X, Y ) or simply (X, Y ); i.e., if f ∈ (X, Y ), then X is 1 (i) Natural isomorphism in group theory. Proc. Natl. Acad. Sci. USA 28, 537Ð544 (1942) (ii) General theory of natural equivalence. Trans. Am. Math. Soc. 58, 231Ð294 (1945) M.R. Adhikari, A. Adhikari, Basic Modern Algebra with Applications, 605 DOI 10.1007/978-81-322-1599-8, © Springer India 2014 606 B Category Theory called the domain of f and Y is called the codomain (or range) of f : one also f writes f : X → Y or X −→ Y to denote the morphism from X to Y ; (c) for each ordered triple of objects X, Y and Z and a pair of morphisms f : X → Y and g : Y → Z, their composite denoted by gf : X → Z i.e., if f ∈ (X, Y ) and g ∈ (Y, Z), then their composite gf satisfies the following two axioms: (i) associativity:iff ∈ (X, Y ), g ∈ (Y, Z) and h ∈ (Z, W), then h(gf ) = (hg)f ∈ (X, W); (ii) identity: for each object Y in C, there is a morphism 1Y ∈ (Y, Y ) such that if f ∈ (X, Y ), then 1Y f = f and if h ∈ (Y, Z), then h1Y = h. Clearly, 1Y is unique. If the class of objects is a set, the category is said to be small. Example B.1.1 (i) Sets and functions form a category denoted by S. (Here the class of objects is the class of all sets and for sets X and Y,(X,Y) equals the set of functions from X to Y and the composition has the usual meaning, i.e., usual composition of functions.) (ii) Sets and injections (or surjections or bijections) form a category. (iii) Groups and homomorphisms form a category denoted by Grp. (Here the class of objects is the class of all groups and for groups X and Y,(X,Y) equals the set of homomorphisms from X to Y and the composition has the usual meaning.) (iv) Rings and homomorphisms form a category denoted by Ring. (v) Commutative rings and homomorphisms form a category. (vi) R-modules and R-homomorphisms form a category denoted by ModR. (vii) Topological spaces and continuous maps form a category denoted by Top. (viii) Finite sets and functions form a category. (ix) Given a partial ordered set (X, ≤), there is a category C whose objects are the elements of X and such that C(x, x ) is either the singleton consisting of the ordered pair (x, x ) or empty, according to whether x ≤ x or x ≤ x ,1x = (x, x) and (q, r)(x, q) = (x, r) when x ≤ q ≤ r. (Note that C(x, x ) is not a set of functions.) (x) Given a category C, there is an opposite (dual) category C0 whose objects Y 0 are in one-to-one correspondence with the objects Y of C and whose morphisms f 0 : Y 0 → X0 are in one-to-one correspondence with the morphisms f : X → Y and for g : Y → Z in C, f 0g0 is defined by f 0g0 = (gf )0. (xi) If C1 and C2 are categories, their product C1 × C2 is the category whose objects are ordered pairs (Y1,Y2) of objects Y1 in C1 and Y2 in C2 and whose morphisms (X1,X2) → (Y1,Y2) are ordered pairs of morphisms (f1,f2), where f1 : X1 → Y1 in C1 and f2 : X2 → Y2 in C2. Similarly, there is a product of an arbitrary indexed family of categories. (xii) Let C be a category. We define a category C2 in the following way: objects of C2 are the morphisms of C; morphisms of C2 are certain pairs of morphisms of C,forf ∈ C(A, B) and g ∈ C(C, D), the pair (α, β) is a morphism from f to g in C2 iff α : A → C and β : B → D satisfy the commutativity relation: βf = gα. B.2 Special Morphisms 607 Note that as objects of C2 and C × C are different, C2 = C × C. (xiii) Exact sequences of R-modules and R-homomorphisms form a category. Definition B.1.2 A subcategory C ⊂ C is a category such that (a) the objects of C are also objects of C, i.e., ob(C ) ⊆ ob(C); (b) for objects X and Y of C , C (X ,Y ) ⊆ C(X ,Y ); (c) if f : X → Y and g : Y → Z are morphisms of C , their composite in C equals their composite in C. Definition B.1.3 A subcategory C of C is said to be a full subcategory of C iff for objects X and Y in C , C(X ,Y ) = C (X ,Y ). The category in Example B.1.1(ii) is a subcategory of the category in Exam- ple B.1.1(i) and the category in Example B.1.1(viii) is a full subcategory of the category in Example B.1.1(i). Remark The categories in Examples B.1.1(iii)Ð(vii) are not subcategories of the category in Example B.1.1(i), because each object of one of the former categories consists of a set, endowed with an additional structure (hence, different objects in these categories may have the same underlying sets). In category Example B.1.1(ix), the morphisms are not functions and so this cat- egory is not a subcategory of the category in Example B.1.1(i). B.2 Special Morphisms Let C be a category and A,B,C,... objects of C. Definition B.2.1 A morphism f : A → B in C is called a coretraction iff there is a morphism g : B → A in C such that gf = 1A. In this case g is called a left inverse of f and f is called a right inverse of g and A is called a retract of B. Dually we say that f is a retraction iff there is a morphism g : B → A such that fg = 1B in C. In this case g is called a right inverse of f . Definition B.2.2 A two-sided inverse (orsimplyaninverse)off is a morphism which is both a left inverse of f and a right inverse of f . Lemma B.1 If f : A → B in C has a left inverse and a right inverse, they are equal. Proof Let g : B → A be a left inverse of f and g : B → A a right inverse of f , then g f = 1A and fg = 1B .Nowg = g 1B = g (fg ) = (g f)g = 1Ag = g . 608 B Category Theory Definition B.2.3 A morphism f : A → B is called an equivalence (or an isomor- phism) in a category C denoted by f : A ≈ B iff there is a morphism g : B → A which is a two-sided inverse of f . Proposition B.2.1 If f : A → B is a morphism in C such that f is both a retraction and a coretraction, then f is an equivalence. Proof It follows from Definition B.2.1 and Lemma B.1. Remark An equivalence f : A ≈ B has a unique inverse denoted by f −1 : B → A and f −1 is also an equivalence. Definition B.2.4 Two objects A and B in C are said to be equivalent iff there is an equivalence f : A ≈ B in C. Remark As the composite of equivalences is an equivalence, the relation of being equivalent is an equivalence relation in any set of objects of a category C. Definition B.2.5 A morphism α ∈ C(A, B) is called a monomorphism (or monic) iff αf = ag ⇒ f = g for all pairs of morphisms f , g with codomain A and same domain in C. Definition B.2.6 A morphism α ∈ C(A, B) is called an epimorphism (or epic) iff fα = gα ⇒ f = g for all pairs of morphism f , g with domain B and same codomain in C. Remark The notion of an epimorphism is dual to that of a monomorphism in the sense that α is an epimorphism in C iff it is a monomorphism in its dual category C0. A coretraction is necessarily a monomorphism and a retraction is an epimorphism. Thus an isomorphism is both a monomorphism and an epimorphism in C. Proposition B.2.2 If α : A → B is a coretraction and also an epimorphism in C, then it is an isomorphism in C. Proof As α is a coretraction, there exists a morphism β : B → A such that βα = 1A. Then (αβ)α = α(βα) = α1A = α = 1B α. (B.1) Since α is an epimorphism, (B.1) shows that αβ = 1B . Consequently α is both a retraction and a coretraction. Hence α is an isomorphism. B.3 Functors Our main interest in categories is in the maps from one category to another. Those maps which have the natural properties of preserving identities and composites are B.3 Functors 609 called functors. An algebraic representation of topology is a mapping from topology to algebra. Such a representation, formally called a functor, converts a topological problem into an algebraic one. Definition B.3.1 Let C and D be categories. A covariant functor (or contravariant functor) T from C to D consists of (i) an object function which assigns to every object X of C an object T(X)of D; and (ii) a morphism function which assigns to every morphism f : X → Y in C,amor- phism T(f): T(X)→ T(Y)(or T(f): T(Y)→ T(X))inD such that (a) T(1X) = 1T(X); (b) T(gf)= T(g)T(f)(or T(gf)= T(f)T(g))forg : Y → W in C. Example B.3.1 (i) There is a covariant functor from the category of groups and homomorphisms to the category of sets and functions which assigns to every group its underlying set. This functor is called a forgetful functor because it forgets the structure of a group. (ii) Let R be a commutative ring. Given a fixed R-module M0, there is a covari- M0 ant functor πM0 (or contravariant functor π ) from the category of R-modules and R-homomorphisms to itself which assigns to an R-module M the R-module HomR (M0,M) (or HomR(M, M0) and if α : M → N is an R-module homomorphism, : → = then πM0 (α) HomR(M0,M) HomR(M0,N) is defined by πM0 (α)(f ) αf M ∀f ∈ HomR(M0,M) (π 0 (α) : HomR(N, M0) → HomR(M, M0) is defined by M π 0 (α)(f ) = fα ∀f ∈ HomR(N, M0)). (iii) Let C be any category and C ∈ ob(C). Then there is a covariant functor hC : C → S (category of sets and functions) defined by hC(A) = C(C, A) (set of all morphisms from the object C to the object A in C) ∀ objects A ∈ ob(C) and for f : A → B in C, hC(f ) : hC(A) → hC(B) is defined by hC(f )(g) = fg ∀g ∈ hC(A) (the right hand side is the composite of morphisms in C). Its dual functor hC defined in an usual manner is a contravariant functor. (iv) For any category C there is a contravariant functor to its opposite category C0 which assigns to an object X of C the object X0 of C0 and to a morphism f : X → Y in C the morphism f 0 : Y 0 → X0 in C0. Remark A functor from a category C to itself is sometimes called a functor on C. Any contravariant functor on C corresponds to a covariant functor on C0 and vice versa. Thus any functor can be regarded as a covariant (or contravariant) functor on a suitable category. In spite of this, we consider covariant as well as contravariant functors on C. Definition B.3.2 A functor T : C → D is called (i) faithful iff the mapping T : C(A, B) → D(T (A), T (B)) is injective; (ii) full iff the mapping T : C(A, B) → D(T (A), T (B)) is surjective; and (iii) an embedding iff T is faithful and T(A)= T(B)⇒ A = B. 610 B Category Theory Definition B.3.3 A category C is called concrete iff there is a faithful functor T : C → S. B.4 Natural Transformations In some occasions we have to compare functors with each other. We do this by means of suitable maps between functors. Definition B.4.1 Let C and D be categories. Suppose T1 and T2 are functors of the same variance (either both covariant or both contravariant) from C to D.Anatural transformation φ from T1 to T2 is a function from the objects of C to morphisms of D such that for every morphism f : X → Y in C the appropriate one of the following conditions hold: φ(Y)T1(f ) = T2(f )φ(X) (when T1 and T2 are both covariant functors) or φ(X)T1(f ) = T2(f )φ(Y ) (when T1 and T2 are both contravariant functors). Definition B.4.2 Let C and D be categories and T1, T2 be functors of the same variance from C to D.Ifφ is a natural transformation from T1 to T2 such that φ(X) is an equivalence in D for each object X in C, then φ is called a natural equivalence. Example B.4.1 Let R be a commutative ring and Mod be the category of R- modules and R-homomorphism, M and N be objects in Mod. Suppose g : M → N is a morphism in Mod. So by Example B.3.1(ii), πM , πN are both covariant functors and π M , π N are both contravariant functors from Mod to itself. Then there exists a ∗ ∗ natural transformation g : πN → πM , where g (X) : πN (X) → πM (X) is defined ∗ by g (X)(h) = hg for every object X in Mod and for all h ∈ πN (X); and a natural transformation g∗ : πM → πN , where g∗(X) is defined in an analogous manner. ∗ If g is an equivalence in Mod, then both the natural transformations g∗ and g are natural equivalences. Theorem B.4.1 (Yoneda’s Lemma) Let C be any category and T a covariant func- tor from C to S (category of sets and functions). Then for any object C in C, there is an equivalence θ = θC,T : (hC,T)→ T(C), where (hC,T)is the class of natural transformations from the set valued functor hC to the set valued functor T such that θ is natural in C and T . B.4 Natural Transformations 611 Fig. B.1 Commutativity of η(C) the rectangle for natural h (C) T(C) transformation η C hC (f ) T(f) η(X) hC(X) T(X) Fig. B.2 Commutativity of hC (g) the rectangle for natural h (X) h (Y ) transformation ρ C C ρ(x)(X) ρ(x)(Y ) T(g) T(X) T(Y) Proof We claim that every object C in C, (hC,T) is a set. Let η : hC → T be a natural transformation. Then for each f : C → X in C the diagram in Fig. B.1 is commutative. Hence T(f) η(C) (1C) = T(f)η(C) (1C) = η(X)hC(f )(1C) = η(X)(f ) (B.2) since hC(1C) = f 1C = f (see Example B.3.1(iii)). This shows that for each X, the function η(X) is completely determined by the element η(C)(1C) ∈ T(C). The latter being a set, so is (hC,T). Having dealt with this part we proceed with the main part. We define θ : (hC,T)→ T(C) by θ(η)= η(C)(1C) ∈ T(C) ∀η ∈ (hC,T). (B.3) We now define a function ρ : T(C)→ (hC,T) by ρ(x)(X)(f ) = T (f )(x) (B.4) ∀x ∈ T(C),X∈ C and f ∈ C(C, X). Then ρ(x) ∈ (hC,T). Now from the definition of ρ it follows that ρ(x)(X) : C(C, X) → T(X) is a function in S, because for f ∈ C(C, X), T(f)∈ S(T (C), T(X))⇒ T (f )(x) ∈ T(X)∀x ∈ T(C). Now for g : X → Y in C, the diagram in Fig. B.2 is commutative. This is because ρ(x)(Y )hC (g)(f ) = ρ(x)(Y )(gf ) = T (gf )(x) = T(g) T (f )(x) = T (g)ρ(x)(X)(f ) ∀f ∈ hC(X). Consequently, ρ(x) is a natural transformation from hC to T . 612 B Category Theory Fig. B.3 Commutativity of θ=θ the rectangle for naturality of C,T θ in T (hC,T) T(C) N∗(α) α(C) θ=θC,S (hC,S) S(C) Fig. B.4 Commutativity of θ =θ the rectangle for naturality of C C,T θ in C (hC,T) T(C) N∗(f ) T(f) θD=θD,T (hD,T) T(D) Now (ρθ)(η) = ρ θ(η) = ρ η(C)(1C) ⇒ (ρθ)(η)(X)(f ) = ρ η(C)(1C) (X)(f ) = Tf η(C)(1C) (B.5) by (B.4) = η(X)(f ) by (B.2) ∀X ∈ C and ∀f ∈ C(C, X) ⇒ ρθ = identity. Also, ∀x ∈ T (C), (θρ)(x) = θ ρ(x) = ρ(x)(C)(1C) (B.6) by (B.3) = T(1C)(X) by (B.4) = 1T(C)(x) = x ⇒ θρ = identity. Consequently, θ is an equivalence. To show that θ is natural in T , we have to prove that the diagram in Fig. B.3 is commutative, where α : T → S is any natural transformation from the set val- ued functor T to the set valued functor S and N∗(α) : (hC,T) → (hC,S) is defined by N∗(α)(η) = αη, the latter is given by (αη)(X) = α(X)η(X) ∀X ∈ C.Nowα(C)θ(η) = α(C)η(C)(1C ) = (αη)(C)(1C ) and θN∗(α)(η) = θ(αη) = (αη)(C)(1C ) ⇒ the above diagram commutes ⇒ θ is natural in T . To show that θ is natural in C, we have to prove that the diagram in Fig. B.4 is commutative. For every morphism f : C → D in C, where for each X ∈ C and η ∈ (ηC,T), N∗(f )(η)(X) : hD(X) → T(X) is defined by N∗(f ) η(X) (g) = η(X)(gf ) ∀g ∈ hD(X). Chasing the element η ∈ (hC,T)anticlockwise, we have θDN∗(f )(η) = N∗(f )(η)(D) (1D) = η(D)(f ) (B.7) B.4 Natural Transformations 613 Fig. B.5 Commutativity of η(C) the rectangle for naturality h (C) T(C) of η C hC (f ) T(f) η(D) hC(D) T(D) and chasing η clockwise, we have T(f)θC(η) = T(f)η(C)(1C ). (B.8) Again by the naturality of η, the diagram in Fig. B.5 is commutative. Hence T(f)η(C)(1C ) = η(D)(hCf)(1C) = η(D)(f 1C) = η(D)(f ) (B.9) Hence (B.7)Ð(B.9) show that θ is natural in T . Remark For the dual result of the theorem, see Exercise 9. Example B.4.2 Let Grp be the category of groups and homomorphisms S be the category of sets and functions and S : Grp → S be the forgetful functor which as- signs to each group G its underlying set SG. Then (i) there is a natural equivalence from the covariant functor hZ to the covariant functor S; and (ii) there is an equivalence θ : (S, S) → SZ. [Hint. (i) Let G be an arbitrary object in Grp and η : hZ → S a natural transfor- mation. Define η(G) : hZ(G) → SG by η(G)(f ) = f(1) ∀f ∈ ηZ(G) = Hom(Z,G), where 1 is the generator of the infinite cyclic group Z. Again define ρ(G) : SG → hZ(G) by ρ(G)(x) = f , where f is the group ho- momorphism : Z → G defined by f(1) = x. Then ρ(G)η(G)(f ) = ρ(G)(f (1)) = ρ(G)(x) = f and η(G)ρ(G)(x) = η(G)(f ) = f(1) = x ⇒ η(G) is an equivalence ∀G ∈ Grp ⇒ η is a natural equiva- lence. (ii) Take in particular C = Grp, C = Z, T = S in Yoneda’s Lemma. Then it fol- ∼ ∼ ∼ lows by (i) and Yoneda’s Lemma that (S, S) = (hZ,hZ) = hZ(Z) = Hom(Z, Z) = Z, the last equivalence is obtained by assigning to the group homomorphism f ∈ Hom(Z, Z) the integer f(1) ∈ Z.] 614 B Category Theory B.5 Presheaf and Sheaf Let X be a topological space. A presheaf A (of abelian groups) on X is a contravari- ant functor from the category of open subsets of X and inclusions to the category of abelian groups and homomorphisms. In general, one may define presheaf with val- ues in an arbitrary category. Thus, if each A(U) is a ring for every open set U ⊂ X and for each pair of open sets of U, V(U⊂ V), ρU,V : A(V ) → A(U) is a ring homomorphism (called restriction) such that ρU,U = identity and ρU,V ρV,W = ρU,W when U ⊂ V ⊂ W ⊂ X, then A is called a presheaf of rings. Similarly, let A be a presheaf of rings on X and suppose that B is a presheaf on X such that each B(U) is an A(U)-module and ρU,V : B(V ) → B(U) are module homomorphisms such that ρU,U = identity and ρU,V ρV,W = ρU,W for open sets U ⊂ V ⊂ W ⊂ X. Then B is said to be a presheaf of A-modules. If M is an abelian group, then there is the ‘constant presheaf’ with A(U) = M for all U and ρU,V = identity ∀U ⊂ V . We also have the presheaf B assigning to U the group (under pointwise addition) B(U) of all functions from U to M, where ρU,V is the canonical restriction.IfM is the group of all real numbers, we have the presheaf C with C(U) being the group of all continuous real valued functions on U. A sheaf (of abelian groups) on a topological space X is a pair A = (S, π), where (i) S is a topological space; (ii) π : S → X is a local homeomorphism, i.e., every point α ∈ S has an open neighborhood N in S such that π|N is a homeomorphism between N and an open neighborhood of π(α) in X; −1 (iii) Each Ax = π (x),forx ∈ X, is an abelian group (and is called the Stalk of S over x); (iv) The group operations are continuous. The meaning of (iv) is as follows: Let S ⊕ S ={(α, β) ∈ S × S : π(α) = π(β)} be the subset of S × S with induced topology, the map S × S → S defined by (α, β) → (α − β) is continuous (equivalently, the map f : S → S, α →−α is continuous and the map S × S → S, (α, β) → (α + β) is continuous). Similarly, we may, for example, define a sheaf of rings or a module (sheaf of modules) over a sheaf of rings. Thus for a sheaf of rings, each stalk is assumed to have the (given) structure of a ring and the map S ⊕S → S, (α, β) → αβ is assumed to be continuous (in addition to (iv)). For a sheaf of R-module each stalk of S is an R-module and the module multiplication S → S defined by α → rα is continuous for each r ∈ R. The Canonical Presheaf of a Sheaf Let A = (S, π) be a sheaf on X. A section of A over an open set U ⊂ X is a continuous map s : U → S such that πs : U → U is the identity on U. By (iii) the set of all sections of A over U is an abelian group denoted by A(U). Similarly, if R is a sheaf of rings, R((U)) is a ring. B.5 Presheaf and Sheaf 615 Fig. B.6 Commutativity of h(U) the rectangle for a A(U) B(U) homomorphism of presheaves ρU,V τU,V h(V ) A(V ) B(V ) Now we assign to each open set U of X the group A(U) of sections of the sheaf A over U, where A(U) is understood to be the zero group if U =∅.IfV ⊂ U, define : A → A ρU,V (U) (V ) (B.10) to be the homomorphism which assigns to each section of A over U, its restriction =∅ = to V (if V , we put ρU,V 0). The assignment → A ⊂ {A U (U) for open sets U X defines a presheaf (U), ρU,V } on X.This presheaf is called the canonical presheaf of the sheaf A = (S, π) on the presheaf of sections of A. The Sheaf Generated by a Presheaf Let A be a presheaf on X. For each open set U ⊂ X, consider the space U × A(U), where U has the subspace topology and = × A(U) has the discrete topology. Form the disjoint union E V ⊂X (U A(U)). We consider the following equivalence relation ρ on E. If (x, s) ∈ U × A(U) and (y, t) ∈ V × A(V ), then (x, s)ρ(y, t) ⇔ (x = y and ∃ an open neighborhood W of x with W ⊂ U ∩ V and ρU,W (s) = ρV,W(t)). Let A be the quotient space E/ρ and π : A → X be the projection induced by the map p : E → X, (x, s) → x. Then π is a local homeomorphism. Clearly, A = π −1(x) is the direct limit of A(U) for U ranging over the open neighbor- hoods of x. Thus the stalk Ax has a natural group structure. Clearly, the group operations in A are continuous (since they are in E). Thus A is a sheaf called the sheaf generated by the presheaf A. Homomorphisms of Presheaves and Sheaves We mainly consider presheaves or sheaves over a fixed base space X.Ahomomorphism h : A → B of presheaves is a collection of homomorphisms hU : A(U) → B(U) commuting the restrictions ⊂ ⊂ i.e., making the diagram in Fig. B.6 commutative for V U X. ρU,V and τU,V are defined by (B.10)inFigB.6. Then h is a natural transformation of functors. A homomorphism f : A → A of sheaves A = (Y, π) and A = (Y ,π ) on X : → A ⊂ A ∀ ∈ is a continuous map f Y Y such that f( x) x x X and the restriction : A → A ∀ ∈ fx x x of f to stalks is a homomorphism x X. A homomorphism of sheaves induces a homomorphism of their corresponding canonical presheaves. Conversely, let h : A → B be a homomorphism of the presheaves. For each x ∈ X, h induces a homomorphism hx : Ax = lim x∈U A(U) → lim x∈U B(U) = Bx , and therefore, a map η : A → B.Ifs ∈ A(U), then h maps the section θ(s)∈ A(U) onto the section θ(h(s))∈ B(U). 616 B Category Theory B.6 Exercises Exercises 1. If α : A → B is a retraction and also a monomorphism in a category C, prove that α is an isomorphism (Dual result of Proposition B.2.2). [Hint. α is a retraction ⇒∃a monomorphism β : B → A such that αβ = 1B . Again α(βα) = (αβ)α = 1B α = α1A ⇒ βα = 1A (as α is a monomorphism).] 2. Show that if α : A → B is an epimorphism in S, then α0 ∈ S0 is a monomor- phism. 3. (a) Let T be a functor from a category C to a category D. Show that T maps equivalences in S to equivalences in D. (b) Show that the equivalences in the category (i) S are bijections; (ii) Grp are isomorphisms of groups; (iii) Ring are isomorphisms of rings; (iv) ModR are isomorphisms of modules; (v) Top are homeomorphisms of topological spaces. 4. Let A be a subspace of a topological space X and f : A → Y continuous. Then f is said to have a continuous extension F : X → Y iff F ◦ i = f , where i : A→ X is the inclusion map. Let T be a covariant (or contravariant) functor from the category of topo- logical spaces and continuous maps to a category S. Show that a necessary condition that a map f : A → Y be extendable to X is that ∃ a morphism φ : T(X)→ T(Y) or φ : T(Y)→ T(X) in S such that φ ◦ T(i)= T(f) or T(f)= T(i)◦ φ . n = 2 n = =[ ∈ n : 5. Let R be the Euclidean n-space, with x xi , E n-ball x R x≤1} and Sn−1 = (n − 1)-sphere ={x ∈ Rn :x=1}. Prove the following: n n (a) the identity map 1Sn : S → S cannot be extended to a continuous map En+1 → Sn; (b) Brouwer Fixed Point Theorem Any continuous map f : En+1 → En+1 has a fixed point (i.e., f(x)= x for some x ∈ En+1). [Hint. (a) If possible, let f : En+1 → Sn be a continuous extension of n ◦ = n 1S . Then f i 1S (see Exercise 4). Assume that Hn is the homology n ∼ n+1 functor such that Hn(S ) = Z and Hn(E ) = 0. ◦ = n n H−→n(i) n+1 H−→n(f ) n Use Hn on f i 1S and obtain Hn(S ) Hn(E ) Hn(S ) such that the composite homomorphism is the identity homomorphism of n Hn(S ) which is not possible as the composite homomorphism Z → 0 → Z cannot be identity. For n = 1, the result (a) can be proved in a similar B.6 Exercises 617 1 ∼ 2 way by using the result that π1(S ) = Z and π1(E ) = 0 (see Sect. 2.10 of Chap. 2). (b) Let f : En+1 → En+1 be a continuous map. If possible f has no fixed point (i.e. f(x)= x for any x ∈ En+1). Then for any x ∈ En+1 join f(x) to x by a line and move along the line in the direction from f(x) to x until the unique point r(x) ∈ Sn is reached. Then r : En+1 → Sn is a continuous map extending 1Sn, which contradicts the result of (a). The Brouwer fixed point theorem for dimension 2 as stated in (b) can be proved 1 ∼ 2 in a similar way by using the result that π1(S ) = Z and π1(E ) = 0(see Sect. 2.10 of Chap. 2).] 6. Let S be a category and (X, Y ) be the set of morphisms from X to Y in S. By keeping X fixed and varying Y , show that this set is an invariant of the equivalent sets in the sense that there is a bijective correspondence between the sets corresponding to the equivalent sets. Find the corresponding result when X is varied and Y is kept fixed. [Hint. Let Y ≈ Z. So there exist f : Y → Z and g : Z → Y such that g ◦ f = 1Y and f ◦ g = 1Z. Then f∗ : (X, Y ) → (X, Z) defined by f∗(α) = f ◦ α and g∗ : (X, Z) → (X, Y ) defined by g∗(β) = g ◦ β are such that (g ◦ f)∗ = g∗ ◦ f∗ = identity and f∗ ◦ g∗ = identity. Consequently f∗ is a bijection.] 7. Let X and Y be objects of a category S and let g : X → Y be a morphism in S. Show that there is a natural transformation g∗ from the covariant functor hY to the covariant functor hX and a natural transformation g∗ from the con- travariant functor hX to the contravariant functor hY . Further show that if g is ∗ an equivalence in S, both these natural transformations g∗ and g are natural equivalences. 8. Let A and C be objects of a category C. Using the Yoneda Lemma, show that (h ,h ) ≈ C(A, C). C A ∼ [Hint. Take T = hA. Then by Yoneda’s Lemma (hC,hA) = hA(C) = C(A, C).] 9. Prove the dual result of Theorem B.4.1: Let C be any category and T a contravariant functor from C to S. Then for any object C ∈ C, there is an equivalence θ : (hC,T)→ T(C) such that θ is natural in C and T . 10. Let Htp denote the category of pointed topological spaces and homotopy classes of their base point preserving continuous maps and Grp be the category of groups and their homomorphisms. (a) If P is an H -group, show that there exists a contravariant function πP from Htp to Grp. Further show that a homomorphism α : P → P between H - groups induces a natural transformation α∗ : π P → π P . (b) If P is a topological group, show that π P is a contravariant functor from HtptoGrp. (c) π1 is a covariant functor from Htp to Grp. Proof (a) Using Theorems 2.9.1Ð2.9.3, Chap. 2, define π P : Htp → Grp, such that π P (X) =[X; P ] and also for f : X → Y in Htp, π P (f ) = f ∗ :[Y ; P ]→ [X; P ] by f ∗[h]=[h ◦ f ], then πP is a contravariant functor. 618 B Category Theory ∗ ∗ ∗ Again, for [g]∈[Y ; P ],f α∗[g]=f [α ◦ g]=[(α ◦ g)◦ f ] and α∗f [g]= ∗ ∗ α∗[g ◦ f ]=[α ◦ (g ◦ f)]⇒f ◦ α∗ = α∗ ◦ f . Consequently, α∗ is a natural transformation. (b) If g1,g2 : X → P are base point preserving continuous maps, then g1g2 : X → P is defined by (g1g2)(x) = g1(x)g2(x) ∀x ∈ X, where the right hand side is the group product in P . The law of composition carries over to give an operation on homotopy classes such that [g1][g2]=[g1g2]. Then [X; P ] is a group. Hence (b) follows from (a). (c) It follows from Sect. 2.10 of Chap. 2. 11. Let C(X) be the ring of real valued continuous functions on a topological space X. Then C is a contravariant functor from the category Top of topological spaces and continuous functions to the category Ring of rings and homomor- phisms. 12. Let Spec (R) be the spectrum space of a ring R endowed with Zariski topology (see Sect. 9.12). Then Spec is a contravariant functor from Ring to Top. 13. (a) Show that all chain complexes and chain maps (see Definition 9.11.4)form a category. We denote this category by Comp. (b) For each n ∈ Z, show that Hn : Comp → Mod is a covariant functor (see Definition 9.11.3). (c) For each n ∈ Z, show that H n: Comp → Mod is a contravariant functor (see Definition 9.11.9). [Hint. For (a) and (b) use Proposition 9.11.4.] 14. Let AB denote the category of abelian groups and their homomorphisms. (a) For an abelian group G,letT(G)denote its torsion group. Show that T : Ab → Ab defines a functor if T(f)is defined by T(f)= f |T(G)for every homomorphism f in Ab such that (i) f is a monomorphism in Ab implies that T(f)is also so; (ii) f is an epimorphism in Ab does not always imply that T(f)is also so. (b) Let p be a fixed prime integer. Show that T : Ab → Ab defines a functor, where the object function is defined by T(G)= G/pG and the morphism function T(f)is defined by T(f): G/pG → H/pH, x + pG → f(x)+ pH for every homomorphism f : G → H in Ab such that (i) f is an epimorphism in Ab implies that T(f)is an epimorphism; (ii) f is a monomorphism in Ab does not always imply that T(f) is a monomorphism. B.7 Additional Reading We refer the reader to the books (Adhikari and Adhikari 2003; Mac Lane 1997; Mitchell 1965; Spanier 1966) for further details. References 619 References Adhikari, M.R., Adhikari, A.: Groups, Rings and Modules with Applications, 2nd edn. Universities Press, Hyderabad (2003) Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1997) Eilenberg, S., Mac Lane, S.: Natural isomorphism in group theory. Proc. Natl. Acad. Sci. 28, 537Ð 545 (1942) Eilenberg, S., Mac Lane, S.: Generalized theory of natural equivalences. Trans. Am. Math. Soc. 58, 231Ð294 (1945) Mitchell, B.: Theory of Categories. Academic Press, New York (1965) Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966) Appendix C A Brief Historical Note Modern algebra began with the work of French mathematician E. Galois (1811Ð 1832) who died in a duel at a very young age. He created one of the most important theories in the history of algebra known as “The Theory of Galois”. Before Galois, algebraists had mainly concentrated on the solutions of algebraic equations. Scip- ione dal Ferro, Tartaglia, and Cardano solved cubic equations, and Ferrari solved biquadratic equations completely by radicals. The most important predecessors of Galois are J.L. Lagrange (1736Ð1813), C.F. Gauss (1777Ð1855) and N.H. Abel (1802Ð1829). Linear algebra arose through the study of the theory of solutions of linear equa- tions and analytic geometry. The former contributed the algebraic formulas and the latter geometric images. The concept of vector spaces (linear spaces) became known around 1920. Hermann Weyl gave a formal definition. A study of matrices, determinants and their closely related topics is the main objective of linear alge- bra. Theories of determinants, quadratic forms, linear algebraic equations and linear differential equations were developed in the first third of the 19th century mainly by J.L. Lagrange and C.F. Gauss. Gauss studied in the fifth part of ‘Disquisniones Arithmeticae’ in 1801, the problem of reduction of the quadratic forms with integral coefficients to canonical forms by using invertible substitution of the variables with integral coefficients. A.L. Cauchy considered in his paper “Memoir on Functions” just two values equal in magnitude but opposite in sign under the permutations of the variables contained in them and extended the investigation of C.A. Vandermonde for determinants of small order. He viewed a determinant as a function of n2 variables which he arranged in a square table. Some years later, C.G.J. Jacobi published a se- ries of papers on theory of determinants and quadratic forms. The existing notation denoting a determinant by means of two vertical bars is introduced by Arthur Cay- ley, a British mathematician. His work on analytical geometry of dimension n and that of H. Grassmann in “The Science of Linear Extension” mark a turning point in the evolution of linear algebra. An important role in studying Linear Algebra is played by B. Riemann (1826Ð1866) in his famous paper “On the Hypothesis that lie at the Foundation of Geometry” and by F. Klein in his work on “n-Dimensional Space V and the Related Geometric Notion of Algebra” in 1870. On the other hand, M.R. Adhikari, A. Adhikari, Basic Modern Algebra with Applications, 621 DOI 10.1007/978-81-322-1599-8, © Springer India 2014 622 C A Brief Historical Note Arthur Cayley introduced the concept of matrices in 1855 and published several notes in Crelle’s Journal. The theory of determinants is much older than the theory of matrices. The German Mathematician Leibniz (1646Ð1716) introduced the con- cept of determinants in connection with the solvability of systems of linear equations communicated in a letter to L’Hospital dated April 28, 1693, but the term “determi- nant” was coined by C.F. Gauss in 1801. Cayley is the first mathematician to realize the importance of theory of matrices and laid the foundation of this theory. The progress in linear algebra during the last few decades focuses several techniques like rank-factorization, generalized inverses and singular value decomposition. Vec- tor spaces over finite fields play a very important role in computer science, coding theory, design of experiments, combinatorics etc. Vector spaces over the rational field Q are very important in number theory and design of experiments and linear spaces over the complex field C are essential for the study of eigenvalues. The the- ory of linear spaces over an arbitrary field F is mainly developed with an eye to the model of linear spaces over R. Leonhard Euler (1707Ð1783) He was born in Basel, Switzerland. He spent most of his time in St. Petersburg and Berlin. He joined the St. Petersburg Academy of Sciences in 1727. He went to Berlin in 1741. He returned to St. Petersburg in 1766, where he remained until his death. He is one of the greatest mathemati- cian of the world. He published 886 papers and books. He is the inventor of the Euler “φ-function”. His other important contributions are “Euler’s formula”, “Eu- ler’s theorem”, “Eulerian angle”, “Eulerian number”, “EulerÐLagrange equation” etc. He represented the Königsberg bridge problem (concerning the seven bridges in Königsberg crossing the river Pregel) by a graph in which the areas are points and the bridges are edges. This study of the seven bridges of Königsberg is the begin- ning of combinatorial topology. His fundamental work in different areas makes his presence everywhere in mathematics. He died in 1783. Joseph Louis Lagrange (1736Ð1813) He was born on January 25, 1736 in Turin, Italy. His mathematical contribution to different branches of mathematics including number theory, theory of equations, differential equations, celestial mechanics, and fluid mechanics is of fundamental importance. In 1771, he presented an extremely valuable memoir to the Berlin Academy ‘Réflections sur la théorie algébrique des équations’. In this paper he tried to prescribe a general method of solution for poly- nomials of degree greater than 4. He, however, failed to achieve his goal. But some new concepts introduced in this paper on permutations of roots stimulated his suc- cessors, like Abel and Galois to develop the necessary theory to find a general method of solution. This paper is considered to be one of the main sources from which modern group theory developed. In 1770, he proved his famous Lagrange’s theorem in group theory. His presence is felt everywhere in mathematics. He died on April 10, 1813. Carl Friedrich Gauss (1777Ð1855) He was born on April 30, 1777 in a poor family, in Brunswick, Germany. At the age of 20, he published the first proof of the fundamental theorem of algebra. Lagrange considered special equations, such C A Brief Historical Note 623 as the cyclotomic equation xn − 1 = 0. But he did not go very far. The complete solution of this equation by means of radicals was given by Gauss in 1801 in his book on number theory ‘Disquisitiones Arithmeticae’, which laid the foundation of algebraic number theory. A second edition of his master piece was published in 1870 (15 years after his death). Gauss√ introduced the concept of congruence class Zn of integers modulo n, notation i for −1, the term complex numbers and Gaus- sian integers Z[i] and studied these extensively. He was the first mathematician to study the structures of fields and groups and established their close relationship. He referred to mathematics as ‘Queen of Sciences’. Gauss did not have interest in teaching. He preferred his job as the Director of Observatory at Göttingen. But he accepted students like Dedekind, Dirichlet, Riemann, Eisenstein and Kummer. E.T. Bell remarked “Gauss lives everywhere in mathematics”. He coined the term ‘determinant’ in 1801. He died on February 23, 1855. Augustin Louis Cauchy (1789Ð1857) He was born on August 21, 1789, in Paris, France. He came in touch with his neighbors Laplace and Bertholet in his childhood. He became an engineer in 1810 and started his mathematical research in 1811 with a problem from Lagrange on convex polyhedrons. He loved teaching. He solved the long standing Fermat’s problem on polygon numbers in 1812. He published more than 800 papers and 8 books on mathematics, mathematical physics and celestial mechanics. His work on mathematics covered calculus, complex functions, alge- bra, differential equations, geometry and analysis. The notion of continuity is his contribution. His treatise on the definite integral submitted to the French Academy in 1814 forms a basis of the theory of complex functions. He made a distinction between permutations and substitutions. The n variables written in any order was called a permutation but a passage from one permutation to another written by 2- row notation called a substitution was introduced by him. We now call a substitution a permutation. He introduced the concept of ‘group of substitutions’ which was later called ‘group of permutations’. Cauchy published a sequence of papers on substi- tutions during 1844Ð1846. The concepts of order of an element, a subgroup, and conjugates are found in his papers. He proved a theorem on a group of finite order, now called, Cauchy’s Theorem in his honor. His work on determinants and matri- ces are also important. Cauchy extended the investigation of C.A. Vandermonde for determinants of small order. He viewed a determinant as a function of n2 variables which he arranged in a square table. He died on May 22, 1857. Niels Henrik Abel (1802Ð1829) He was born on August 5, 1802, in Finnöy, Nor- way. While he was a school student, he was greatly influenced by his mathematics teacher Holmbëë to read the work of Euler, Lagrange, Laplace, and Cauchy. In- spired by their work he began to solve the then unsolved problem of solvability of the quintic equation. After a long effort, he proved in 1824 that a solution of this problem by radicals is impossible. He then published a leaflet in French entitled ‘Mémorie sur les équations algébriques’ in 1824. While proving this impossibil- ity, he used some results obtained by Lagrange and Cauchy. The groups in which composition is commutative are now called Abelian in his honour. His work on el- liptic functions revolutionized the theory of elliptic functions. He moved to Paris 624 C A Brief Historical Note and Berlin in search of a teaching assignment. The information of his appointment as a Professor of Mathematics at the University of Berlin reached his home 2 days after his death from tuberculosis on April 6, 1829. An Abel Prize (equivalent to Nobel Prize) has been instituted in 2002 by Norway Government to mark his 200th Birth Anniversary. C.G.J. Jacobi (1804Ð1851) He published a number of papers on theory of de- terminants and quadratic forms. The identity introduced by him, called “The Jacobi Identity” is a relationship [A,[B,C]]+[B,[C,A]]+[C,[A,B]] = 0, between three elements A, B, and C, where [A,B] is the commutator. The elements of a Lie al- gebra satisfy this identity. Jacobi’s Identity has wide applications in science and engineering. H. Grassmann (1809Ð1877) His work on analytical geometry for dimension n appeared in “The Science of Linear Extension” and marked a turning point in the evolution of linear algebra. Evariste Galois (1811Ð1832) He was born on October 25, 1811 in Bour-la-Reine, near Paris, France. Galois twice failed at the entrance examination for the École Polytechnique. Galois presented his first paper on the solution of algebraic equa- tions to the Académie des Sciences de Paris in May, 1829. He also presented his second paper on equations of prime degree to the Academy. Both the papers were sent to Cauchy but were lost for ever. He presented another paper on solution of algebraic equations to the Academy in February, 1830 which was sent to Fourier by the Academy. This paper was also lost for ever. Galois published in the “Bul- letin des Sciences Mathématiques of Férussac” in April 1830 an article wherein he announced some of his main results of the lost papers. One theorem of the paper states “In order that an equation of prime degree is solvable by radicals, it is nec- essary and sufficient that, if two of its roots are known, the others can be expressed rationally. This shows that the general equation of degree 5 cannot be solved by radicals.” He published two more papers in June, 1830 on resolutions of numeri- cal equations and on the structure of finite field. Galois presented to the Academy a new version of his memoir entitled “Memoire sur les conditions de résolubilité des équations par radicaux” in January, 1831. This paper was published in 1846 (14 years after the death of Galois) by Liouville in the Journal “de mathématiques pures at appliqués II”. An edition containing all preserved letters and manuscripts of Ga- lois was published by Gauthier-Villars in 1962 under the title “Ecrits et mémoires mathématiques d’Evariste Galois”. Arthur Cayley (1821Ð1895) He was born on August 16, 1821, in Cambridge, England. He became a lawyer in 1849. During his practice in the legal profession up to 1863, he wrote about 300 mathematical papers. He joined as Professor of Pure Mathematics at Cambridge in 1863 and worked there up to his death on Jan- uary 26, 1895. He worked on mathematics, theoretical dynamics and mathematical astronomy. He wrote 966 papers and one book. In 1854, Cayley published two pa- pers under the title “On the Theory of Groups Depending on the Symbolic Equation θ n = 1” in Philosophical Magazine of Royal Society, London, Vol. 7. In 1878, he C A Brief Historical Note 625 gave the abstract definition of groups and formulated the problem: to find all finite groups of a given order n, and proved the famous Cayley Theorem for any finite group. He introduced multiplication table for a finite group known as Cayley Ta- ble. He introduced the concept of matrices in 1855 and published several notes in Crelle’s Journal. He is the first mathematician who realized the importance of theory of matrices and laid the foundation work on this theory. He developed matrix the- ory and proved CayleyÐHamilton theorem. He is one of the earlier mathematicians who studied geometry of dimensions greater than 3. The existing notation denoting a determinant by means of two vertical bars was introduced by A. Cayley. He is considered as the founder of abstract group theory. Leopold Kronecker (1823Ð1891) He was born on December 7, 1823 in Ger- many. After 1870, the abstract notion of “groups” was developed in several stages, essentially due to Kronecker (1870) and Cayley (1878). The modern definition of a group by axioms was given for abelian groups by Kronecker in 1870. After the introduction of the concept of abstract group by Cayley and Kronecker, the theory of groups changed its character. Earlier to them, the main problem was to determine the structure of permutation groups under certain conditions as well as to determine the structure of finite dimensional continuous groups of transformations. But after- wards, the main problem became: to create a general theory of the structure of ab- stract groups, and to determine all finite groups of a given order. E.E. Kummer was his mathematics teacher. Kronecker was greatly inspired by Kummer for research. Kronecker’s work on algebraic number theory placed him as one of the inventors of algebraic number theory along with Kummer and Dedekind. Kronecker is consid- ered to be the first mathematician who clearly understood the wok of Galois. While Weierstrass and Cantor were creating modern analysis, Kronecker remarked “God made positive integers, all else is due to man”. This remark badly affected Cantor. Kronecker died on December 29, 1891. Bernhard Riemann (1826Ð1866) His famous paper “On the Hypothesis that lie at the Foundation of Geometry” plays an important role in studying linear algebra. His idea on geometry of space has made a significant effect on the development of modern theoretical physics. He clarified the notion of integral by defining an integral called Riemann Integral. Richard Dedekind (1831Ð1916) He was born on October 6, in 1831 in Brunswick, Germany. He was in contact with Gauss, Dirichlet, Riemann. He com- pleted his Ph.D. work under Gauss. He attended a series of lectures of Dirichlet on theory of numbers and lectures of Riemann on Abelian and elliptic functions. He became interested in analytic geometry and algebraic analysis, differential integral calculus and also in mechanics. He introduced the concept of “Dedekind cut” in 1872. He edited the work of Gauss, Dirichlet, Riemann. He coined the term “ideal” and developed ideal theory and unique factorization theory. While studying ideals in algebraic number fields, he introduced the concepts of ascending chain condition. He replaced the concept the permutation group by abstract group. He loved teaching on Galois theory. He died on February 12, 1916. 626 C A Brief Historical Note Peter Ludvig Mejdell Sylow (1832Ð1918) He was born on December 12, 1832 in Oslo, Norway. In 1872, he published a paper of fundamental importance in Math. Annalen 5, 584Ð594. It contained eight theorems and extended Cauchy’s result. The theorems in this paper are known as Sylow’s Theorems. He is famous for his work on structural results in finite group theory. He died on September 7, 1918. Sylow and Lie prepared an edition on the complete work of Abel during 1883Ð1881 with Sylow as the main author according to Lie. Camille Jordan (1838Ð1922) He was born on January 5, 1838 in Lyons, France. He was an engineer but he took admission at École Polytechnique in 1855 to study mathematics. He became professor of analysis in École Polytechnique in 1876. Jor- dan published 120 papers in mathematics. He originated the notion of a bounded function. He proved his famous theorem: “A plane can be decomposed into two re- gions by a simple closed curve” (called Jordan curve theorem). He was greatly influ- enced by the work of Riemann. He used combinatorial methods to work in topology and introduced the concept of path homotopy. Jordan was basically an algebraist. He developed the theory of finite groups and its applications following Galois. He introduced the concept of composition series and proved the famous Jordan–Hölder Theorem and the concept of simple groups and epimorphisms. His mathematical work of 667 pages “Traité des substitutions et des équations algébriques” published in 1870, attracted many scholars like Sophus Lie from Norway and Felix Klein from Germany. In the preface to his “Traité”, he acknowledged the contribution of his pre- decessors: Galois who invented the principles of Galois Theory, Betti who wrote a memoir, in which the complete sequence of Galois Theory had been rigorously es- tablished for the first time, Abel, Kronecker, and Cayley. He proved finiteness the- orems and introduced the concept of simple groups. He studied the general linear groups and the fields of p elements (p is a prime) and applied his work to classical groups to determine the structure of Galois groups of equations. He died on January 22, 1922. Sophus Lie (1842Ð1899) He was born at Nordfjordeid, Norway, in December 1842. He made significant joint work with C.F. Klien. They went to Paris in 1870 and lived in adjacent rooms for 2 months. Their joint paper was published in 1871 in Math. Annalen 4, 424Ð429. They considered one dimensional continuous groups. In later years, they moved in different directions. S. Lie developed his theory of continuous maps and used it in investigating differential equations and C.F. Klein investigated discrete groups. S. Lie is the inventor of Lie Algebra. The fundamental idea of his Lie theory was published in his paper in Math. Annalen 16, in 1880. Lie’s investigation on the integration of differential equations attracted himself to investigate groups of transformations transforming a differential equation into itself. He developed his theory of transformation groups to solve his integration problems. Continuous transformation groups (called Lie groups, named after Lie), commutator brackets and Lie algebras are essentially due to him. They have wide applications in quantum mechanics. He died in 1899. C A Brief Historical Note 627 Georg Ferdinand Ludwig Philipp Cantor (1845Ð1918) He was born in March 1845. He was a German mathematician. He originated naive set theory. He is known as the father of set theory, which forms a strong foundation of different disciplines in modern science. There are two general approaches to set theory. The first one is called “naive set theory” and the second one is called “axiomatic set theory”, also known as “Zermelo–Fraenkel set theory”. These two approaches differ in a number of ways. While developing mathematical theory of sets to study the real numbers and Fourier series, Cantor established the importance of a bijective correspondence between two sets and introduced the concepts of infinite sets, well-ordered sets, cardinal and ordinal numbers of sets and their arithmetic. He proved that the set of real numbers is not countable by a process known as Cantor’s diagonal process. This result shows the existence of transcendental numbers (which are not algebraic over the field Q of rational numbers) as the set A of algebraic numbers is countable. Transcendental numbers are precisely all the members of R of real numbers which are not algebraic over the field Q. He died in January 6, 1918. C. Felix Klein (1849Ð1925) He did not consider groups in his first paper on Non- Euclidean geometry but he considered transformation groups of invertible transfor- mations of a manifold in his second paper published in 1873. He defined group like Jordan. According to Klein, projective geometry and Euclidean geometry deal with properties of figures which are invariant under respective transformations. Klein-4 group is named in honor of Klein. His work on “n-dimensional vector spaces and the related geometric notion of algebra’ in 1870 stimulated the study of linear alge- bra. Inspired by seminar lectures of Kronecker and Klein, the algebraist O.L. Hölder (1859Ð1937) completed the proof of so-called Jordan–Hölder theorem on composi- tion series, which plays a fundamental role in group theory. Jules Henri Poincaré (1854Ð1912) He is a French mathematician and is known as the father of topology. He was born in 1854 in Nancy, Lorraine, France. He has published around 300 papers and several books spread over different areas of pure mathematics, celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity, physical cosmology and philosophy of science. The fundamental group defined in Chap. 3 was invented by Henri Poincaré in 1895 through his work Anal- ysis Situs. This group is also called Poincaré’s group in his honor. This group is associated to any given pointed topological space. It is a topological invariant in the sense that two homeomorphic pointed topological spaces have isomorphic funda- mental groups. Intuitively, this group provides information about the basic shape, or holes, of the topological space. Poincaré’s work in algebraic topology is mainly in geometric terms. The fundamental group is the first and simplest of the homotopy groups. Poincaré is the first mathematician who applied algebraic objects in homo- topy theory. Historically, an idea of a fundamental group was in the study of Rie- mann surfaces by Bernhard Riemann, Henri Poincaré, and Felix Klein. This group has wide applications in different areas. For example, Brouwer fixed point theo- rem and fundamental theorem of algebra are proved in this book and an extension 628 C A Brief Historical Note problem is solved by using homotopy theory and this group. His result on index of intersection of two subgroups of a finite group is known as Poincaré’s theorem. He died in 1912 in Paris, France. His famous Poincaré conjecture, which was one of the most important long standing unsolved problems in mathematics till 2002Ð2003. The conjecture is “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.” Grigory Perelman, a Russian mathematician solved this problem. He was awarded a Fields Medal at the Madrid, Spain meeting of the International Congress of Mathematicians (ICM) for “his contributions to geometry and his revo- lutionary insights into the analytical and geometric structure of the Ricci flow”. But he declined to accept it. David Hilbert (1862Ð1943) He was born on January 23, 1862 in Königsberg, Germany. The great mathematician Lindeman stimulated Hilbert to work on the theory of invariants. He proved his famous theorem known as Hilbert Basis the- orem which made a revolution in algebraic geometry and ring theory. Greatly in- fluenced by the axioms of Euclid, he proposed 21 axioms with their significance. While addressing the Second International Congress of Mathematicians (ICM) of 1900 at Paris, Hilbert posed 23 interesting problems for investigation including the continuum hypothesis, well ordering of reals, transcendence of powers of algebraic numbers, Riemann hypothesis, extension of the Principle of Dirichlet and others. He also worked on algebraic number theory, foundation of geometry, integral equa- tion, calculus of variations, functional analysis and theoretical physics. He died on February 14, 1943. Amalie Emmy Noether (1882Ð1935) She was born on March 23, 1882 in Erlan- gen, Germany. Her father, Max Noether was a famous mathematician. She started her mathematics career at the University of Göttingen in 1903 as a non-regular stu- dent, as at that time girl students were not allowed to be admitted as regular students. However, she was permitted in 1904 to enroll at the University of Erlangen where her father taught. Hilbert invited her to Göttingen in 1915. After prolonged efforts of Hilbert, she was appointed an associate professor and taught there from 1922 to 1933. As she was a Jew she had to leave the university and went to USA in 1933 because of the rise of the Nazi regime. She gave lectures and did research work at Bryn Mawr College. In 1921 Noether extended the Dedekind theory of ideals and the representation theory of integral domains and rings of algebraic numbers for arbitrary commutative rings satisfying ascending chain condition. These rings are now called Noetherian ring. Motivated by Hilbert’s axiomatization of Euclidean geometry, Noether became interested in an abstract axiomatic approach to ring the- ory. Noether developed a general representation theory of groups and algebras over arbitrary ground fields. Her 45 research papers are divided into four categories: Category 1. Group-Theoretic Foundations. Category 2. Non-commutative Ideal Theory. Category 3. Modules and Representations. Category 4. Representations of Groups and Algebras. C A Brief Historical Note 629 The basic concept of modern theory of rings came through the works of Noether and Artin during 1920s. The ascending chain condition was introduced by Noether in her theory of ideals. The concept of Noetherian rings is her invention. She died on April 14, 1935. Joseph Henry MecLagan Wedderburn (1882Ð1948) He was born on Febru- ary 26, 1882 in Forfar, Scotland. He published 38 papers from 1905 to 1928 and a book on matrices in 1934. He also worked on the structure of algebras over arbitrary fields, instead of algebras over the fields of complex numbers or real numbers. He proved the celebrated theorem in 1905 known as ‘Wed- derburn theorem’ which asserts that a finite division ring is commutative. Its intrinsic beauty is for interlinking the number of elements in a certain alge- braic system and the multiplication of that system. This result appears in many contexts and developed a large area of research. His work on finite algebra made a revolution in projective geometry with a finite number of points. For example, his proof of the geometrical result that Desargues configuration im- plies Pappus configuration is of immense intrinsic beauty. He died on October 9, 1948. Emil Artin (1898Ð1962) He was born on March 3, 1898 in Vienna, Austria. Artin generalized some results of Wedderburn on algebras over fields in 1927, by considering rings with descending chain condition. He worked on various areas of mathematics such as number theory, group theory, ring theory, field theory, geomet- ric algebra, and algebraic topology. Artinian ring is his invention. He proved in 1927 the general laws of reciprocity, which cover all the previous laws of reciprocity up to the time of Gauss. He formulated Galois theory in an abstract setting, he published Galois theory, as used today by establishing a connection between field extension and the subgroups of automorphisms. He died on December 20, 1962. Oscar Zariski (1899Ð1986) He was born in the city of Kobrin, the then part of the Russian Empire. In 1920, the city Kobrin fell in independent Poland as per po- litical agreement between Russia and Poland. He opted for the Polish nationality for his convenience to study mathematics. His work in the area of algebraic geom- etry is fundamental. The subject algebraic geometry arose through the study of the solution sets of polynomial equations and can be traced back to Descartes. But it becomes an important mathematical discipline in the 19th and 20th centuries. By the early twentieth century, the Italian school of algebraic geometers established many interesting results and investigated many basic problems of algebraic geome- try. Zariski joined the strong group of the Italian school and by using modern algebra the school reformulated the subject. Zariski introduced a topology with the help of algebraic sets as closed sets. This topology is now known as Zariski topology. He was a professor of mathematics at Harvard University and made Harvard a world center for algebraic geometry and formed the basis for its twentieth century devel- opment. 630 C A Brief Historical Note C.1 Additional Reading We refer the reader to the books (Adhikari and Adhikari 2003, 2004; Bell 1962; Birkoff and Mac Lane 2003; Hazewinkel et al. 2011; van der Waerden 1960) for further details. References Adhikari, M.R., Adhikari, A.: Groups, Rings and Modules with Applications, 2nd edn. Universities Press, Hyderabad (2003) Adhikari, M.R., Adhikari, A.: Text Book of Linear Algebra: An Introduction to Modern Algebra. Allied Publishers, New Delhi (2004) Bell, E.T.: Men of Mathematics. Simon and Schuster, New York (1962) Birkhoff, G., Mac Lane, S.: A Survey of Modern Algebra. Universities Press, Andhra Pradesh (2003) Hazewinkel, M., Gubareni, N., and Kirichenko, V.V.: Algebras, Rings and Modules, Vol. 1. Springer, New Delhi (2011) van der Waerden, B.L.: A History of Algebra. Springer, Berlin (1960) Index Symbols Algebra of vectors, 274 F -isomorphism, 491 Algebra over a field, 300 F -liar, 469 Algebraic closure, 512 F -witness, 469 Algebraic element, 489 G-equivariant map, 142 Algebraic extension, 489 G-homomorphism, 142 Algebraic integer, 504 G-set, 138 Algebraic multiplicity, 315 G-space, 151 Algebraic number, 500 H -group, 114 Algebraic number field, 502, 509 I : J , 231 Algebraic set, 214 n-boundaries, 111 Algebraic set of zeros, 214 n-chains, 406 Algebraically closed field, 502 n-cycles, 111 Alternating group, 127 O-notation, 449 Amicable number, 414 p-groups, 144 André Weil, 214 ∅, 3 Annihilator, 365 Antisymmetric, 9 A Antisymmetric relation, 9 Abel, 623 Arithmetic function, 420 Abelian group, 73 Artin, 629 Access structure, 548 Artinian module, 392 Action of a complex Lie group, 152 Artinian ring, 260 Action of a group, 137 Ascending (descending) chain condition, 258 Action of a real Lie group, 152 Associates, 238 Action of semigroup, 155 Associative algebra, 358 Actions of topological group, 151 Associative law, 22 Additivity law, 290 Asymmetric cryptosystem, 529 Affine algebraic set, 214 Asymptotic notation, 448 Affine algebraic surface, 214 Affine cipher, 523 B Affine combination, 289 Banach space, 331 Affine scheme, 410 Basis, 284 Affine set, 288 Basis of a module, 375 Affine variety, 217 Basis of a vector space, 284 Affinely independent, 289 Betti number, 383 AKS Algorithm, 465 Bezout’s identity, 241 Algebra ideal, 301 Bidding attack, 535 M.R. Adhikari, A. Adhikari, Basic Modern Algebra with Applications, 631 DOI 10.1007/978-81-322-1599-8, © Springer India 2014 632 Index Bijection, 21 Commutative ring, 160 Bijective, 21 Companion matrix, 345 Bilinearity, 327 Complement, 5 Binary operation, 56 Complement of a subspace, 278 Binary relation, 8 Complete field, 502 Boole, 161 Complete metric space, 331 Boolean algebra, 38 Complex Lie group, 152 Boolean ring, 193 Complex projective space, 152 Boundary homomorphism, 406 Complex vector space, 275 Brower fixed point theorem, 616 Composition, 56 Brute-force search, 521 Concatenation of words, 156 Burnside theorem, 143 Congruence class of matrices, 336 Congruence classes, 48 C Congruence modulo, 12 Caesarian cipher, 522 Congruent, 48 Canonical homomorphism, 279 Congruent matrices, 336 Cantor, 627 Conjugacy, 84 Cardinal number, 27 Conjugacy class, 84 Carmichael number, 472 Conjugate, 84 Cartesian product, 6 Conjugate bilinearity, 327 Category, 605 Conjugate element, 138 Cauchy, 623 Conjugate matrix, 314 Cauchy sequence, 331 Conjugate symmetry, 326 CauchyÐSchwarz inequality, 327 Conjugate transpose, 314 Cauchy’s theorem, 144 Conjugation, 138 Cayley, 624 Continuum hypothesis, 31 Cayley table, 56 Converse of Lagrange’s theorem, 145 CayleyÐHamilton theorem, 317 Coordinate ring, 217 Cayley’s theorem, 140 Coretraction, 607 Center of a ring, 167 Correspondence theorem for modules, 370 Chain complex, 111, 406 Correspondence theorem for rings, 208 Chain homotopic, 407 Countability of a set, 25 Chain map, 111 Countable set, 25 Characteristic equation, 311 Cryptanalysis, 519 Characteristic of a ring, 169 Cryptology, 519 Characteristic polynomial, 311 Cryptosystem, 518 Characteristic root, 310 Cyclic module, 361, 376 Chinese remainder theorem for modules, 405 Cyclic semigroup, 64 Chosen cipher text attack, 521 Cyclotomic polynomial, 250 Chosen plain text attack, 521 Cipher text, 518 D Cipher text only attack, 520 Decision problem, 451 Class equation, 146 Decryption, 518 Classical canonical form, 323 Dedekind, 625 Closure axiom, 56 Degree of an element, 490 Coboundary homomorphism, 408 Desargues, 629 Cochain complex, 408 Determinant divisor, 320 Cochain map, 409 Determinant function, 98 Cohen’s theorem, 262 Deterministic algorithm, 451 Cohomology class, 408 Deterministic primality testing, 463 Cohomology group, 408, 111 Diagonal, 6, 338 Cohomology module, 408 Diamond lattice, 38 Column rank, 308 Difference group, 94 Commutative, 73 Differential operator, 340 Index 633 Diffie-Hellman key exchange, 530 Faithful, 394 Diffie-Hellman problem (DHP), 532 Faithful functor, 609 Digital signature, 542 Faithful module, 394 Dihedral group, 99 Family of sets, 7 Dimension, 286 Fermat number, 443 Direct product of modules, 363 Fermat prime, 443 Direct sum, 278 Fermat’s last theorem, 414 Direct sum of the modules, 364 Fermat’s little theorem, 425 Dirichlet, 21 Field, 165 Dirichlet’s theorem on primes, 418 Field extension, 487 Discrete logarithm problem (DLP), 530 Field generated, 488 Divide, 237 Field of formal power series, 181 Division algorithm, 42 Field of quotients, 174 Division algorithm in polynomial ring, 253 Field of rational functions, 182 Division ring, 163 Finite extension, 492 Domain, 19 Finite field, 498 Dorroh extension theorem, 173 Finite set, 25 Dual space, 297 Finitely generated, 85, 204, 281 Duplication of a cube, 513 Finitely generated module, 361 First isomorphism theorem for modules, 371 E First isomorphism theorem for vector spaces, Eigenspace, 311 294 Eigenvalue, 310 First Sylow theorem, 147 Eigenvector, 310 Fitting’s lemma, 394 Elementary divisor, 321 Five lemma, 390 Elementary divisors for modules, 384 Formal power series ring, 177 ElGamal cryptosystem, 536 Four lemma, 389 Embedding in a ring, 172 Fourier coefficient, 328 Encryption, 518 Free abelian group, 103 Endomorphism ring, 186 Free action, 139, 151 Epimorphism theorem for modules, 371 Free module, 376 Equivalence class, 11 Free part of module, 383 Equivalence relation, 8 Free rank, 383 Equivalent matrices, 307 Frequency analysis, 524 Euclidean algorithm, 457 Frobenius homomorphism, 172 Euclidean domain, 243 Full linear group, 308 Euclidean norm function, 243 Function, 19 Euclidean valuation, 243 Functor, 608 Euclid’s theorem, 46 Fundamental group, 115 Euler, 622 Fundamental homomorphism theorem for Euler phi-function, 421 modules, 371 Euler’s theorem, 425 Fundamental theorem of algebra, 502 Exact sequence, 386 Fundamental theorem of arithmetic, 46 Existence theorem of a basis, 284 Exponent of a module, 383 G Exponential time solvable, 452 Galois, 624 Extended Euclidean algorithm, 460 Galois field, 498 Extension of a ring, 172 Gauss, 622 External law of composition, 274 Gauss lemma, 434 Gauss theorem, 248 F Gaussian integer, 506 Factorisable, 46 Gaussian prime, 506 Factorization, 242 Gelfand-Kolmogoroff, 185 Factorization domain, 242 General linear group, 76, 308 634 Index General secret sharing scheme, 548 Inner automorphism, 141 Geometric multiplicity, 315 Inner product, 326 Gödel, 31 Inner product space, 327 GramÐSchmidt orthogonalization process, Instance of a computational problem, 451 328 Integral domain, 162 Grassmann, 624 Integral quaternion, 164 Greatest common divisor, 43, 240 Intermediate field, 494 Grothendieck, 411 Intersection, 4 Group, 73 Invariant factor, 320, 383 Group action, 138 Invariant subspace, 344 Group of units, 424, 425 Inverse, 23 Groupoid, 57 Invertible, 163 Invertible linear operator, 296 H Irrational flow, 154 Hamilton, 164 Irreducible element, 239 Hasse diagram, 13 Isomorphism of field extensions, 491 Hermitian matrix, 314 Isomorphism theorem for modules, 370 Hilbert, 628 Isomorphism theorem for rings, 207 Hilbert basis, 342 Isomorphism theorem for vector spaces, 294 Hilbert basis theorem, 259 Isomorphism theorem of quotient of quotient Hilbert space, 333 for modules, 371 Hilbert’s nullstellensatz theorem, 220 Isotropy, 139 Hill cipher, 526 Isotropy group, 139 Homogeneity law, 290 Isotropy subgroup, 142 Homogeneous G-set, 142 Homogeneous set, 142 J Homology class, 406 Jacobi, 624 Homology group, 111, 406 Jacobi symbol, 440 Homology module, 406 Jacobson radical, 264 Homomorphisms of presheaves, 615 Jordan, 626 Homomorphisms of sheaves, 615 Jordan block, 321 Homothecy, 369 Jordan canonical, 322 Homotopy, 115 Jordan form theorem, 322 Hopf group, 112, 114 Jordan normal form, 323 Hypersurface, 214 K I Kerckhoffs’ law, 519 Ideal, 204 Klein, 627 Ideal of a set, 216 Klein-4 group, 124 Idempotent, 63 Known plain text attack, 521 Identification map, 151 Kronecker, 625 Identity element, 160 Kronecker delta, 298 Incongruent, 48 Krull dimension, 271 Indefinite, 339 Krull-Zorn, 212 Indeterminate, 179 Index, 339 L Index of a subgroup, 140 Lagrange, 622 Indexing set, 7 Lagrange’s theorem, 90 Infinite dimensional Euclidean space, 332 Law of quadratic reciprocity, 438 Infinite dimensional unitary space, 332 Least common multiple, 51, 241 Infinite extension, 492 Least common multiple (lcm), 241 Infinitely countable set, 25 Left ideal, 204 Infinitely generated, 281 Left invertible, 163 Injective, 21 Left translation, 138 Index 635 Legendre symbol, 432 O Leibniz, 622 Oblivious transfer, 544 Lie, 626 OneÐone, 21 Lie group, 113 One-to-one correspondence, 21 Linear combination, 280 One-way, 528 Linear functional, 297 Onto, 21 Linear functionals on a module, 374 Open source software, 565 Linear operator, 290 Orbit, 139 Linear space, 274 Orbit set, 139 Linear transformation, 290 Orbit space, 151 Linearly independent, 282 Order of a power series, 179 Local ring, 213 Ordered pair, 6 Loop, 115 Orthogonal basis, 328 Orthogonal complement, 330 M Orthogonal decomposition, 330 Map, 19 Orthogonal direct sum, 330 Matrix representation, 304 Orthogonal group, 122 Maximal condition, 258 Orthonormal basis, 328 Maximal ideal, 209 OTbasedonRSA,546 Maximal submodule, 362 Mersenne number, 447 P Mersenne prime, 447 Pappus, 629 Metric space, 330 Parallelogram law, 333 Minimal condition, 260 Partial order, 12 Minimal polynomial, 302, 490, 501 Partition, 10 Minimal polynomial of a matrix, 318 Partition of a set, 10 Minimal qualified subset, 548 Path, 115 Minimal submodule, 362 Path connected, 115 Module, 356 Peano’s axiom, 39 Module homomorphism, 366 Pentagonal lattice, 16 Module of homomorphisms, 373 Perfect number, 445 Modules with chain conditions, 392 Perfect secret sharing, 548 Monic polynomial, 182 Permutation group, 75 Mono-alphabetic, 525 Permutation matrix, 304 Monoid, 59 Pigeonhole principle, 21 Monotone access structure, 548 Pixel expansion, 552 Multiplicative function, 420 Plain text, 518 Poincare, 627 N Poly-alphabetic, 525 Nakayama’s lemma, 401 Polynomial ring, 181 Natural transformation, 610 Polynomial time solvable, 452 Negative definite, 339 Polynomials in several variables, 196 Negative semidefinite, 339 Poset, 12 Nil ideal, 265 Positive definite, 339 Nilpotent, 63, 193 Positive definiteness, 326 Noether, 628 Positive semidefinite, 339 Noetherian module, 392 Power series rings, 177 Noetherian ring, 258, 260 Power set, 7 Non-singular linear operator, 296 Presheaf of rings, 614 Normal form, 338 Primary decomposition theorem for modules, Normed linear space, 331 384 Null space, 292 Primary ideal, 230 Nullity, 295 Prime element, 239 Number of divisor function, 423 Prime field, 168, 489 636 Index Prime ideal, 209 Regular semiring, 593 Prime integer, 45 Reimann sphere, 24 Prime number theorem, 419 Remainder theorem, 253 Prime subfield, 177 Residue classes modulo n, 12 Primitive element, 488 Retract, 607 Primitive root, 426 Riemann, 625 Principal ideal, 204 Right ideal, 204 Principal ideal domain (PID), 212 Right invertible, 163 Principle of mathematical induction, 40 Ring, 160 Principle of strong mathematical induction, 47 Ring completion, 596 Principle of well-ordering, 41 Ring embedding, 172 Private key, 528, 527 Ring homomorphism, 170 Probabilistic primality testing, 466 Ring of continuous functions, 185 Product of ideals, 206 Ring of functions, 184, 193 Proper ideal, 205 Row rank, 308 Property of trichotomy, 40 RSA based digital signature, 543 Pseudo-prime, 469 RSA public key cryptosystem, 532 Public key, 528, 529 S Q SAGE, 565 Quadratic domain, 252 Scalar, 275 Quadratic field, 510 Scalar multiplication, 274 Quadratic form, 334 Scheme, 411 Quadratic non-residue, 430 Schroeder-Bernstein equivalence principle, 28 Quadratic residue, 430 Schroeder-Bernstein theorem, 29 Quaternion group, 123 Schur’s lemma, 369 Quaternion rings, 164 Second isomorphism theorem for modules, Quotient algebra, 301 371 Quotient field, 174 Second principle of mathematical induction, Quotient module, 370 47 Quotient ring, 206 Second Sylow theorem, 147 Quotient semigroup, 156 Secret sharing, 547 Quotient set, 11 Semiautomation, 155 Quotient space, 279 Semigroup, 58 Semisimple, 364 R Semisimple ring, 231 Rabin cryptosystem, 538 Set, 2 Randomized primality testing, 466 Set of generators, 281 Range, 19 Set of residue classes, 11 Range space, 292 Shadow, 548 Rank, 295 Shamir’s threshold secret sharing scheme, 549 Rank of a free module, 377 Share, 548 Rank of a matrix, 309 Sheaf, 614 Rational canonical form, 345 Shift cipher, 522 Rational integer, 504 Short exact sequence, 386 Rational quaternion, 164 Short five lemma, 391 Real Lie group, 152 Signature, 339 Real projective space, 151 Similar matrices, 306 Real quaternion, 164 Simple extension, 488 Real quaternion ring, 164 Simple module, 362 Real vector space, 275 Simple ring, 205 Rectangular band, 63 Simplex, 289 Reflexive, 8 Skew field, 163 Regular semigroup, 63 Skew-Hermitian matrix, 314 Index 637 Smith normal form, 343 Torsion module, 383 Solution space, 276 Torus, 152 Space curve, 214 Transcendental element, 489 Space spanned, 281 Transcendental extension, 489 Special morphisms, 607 Transcendental number, 500 Special quaternions, 165 Transfinite cardinal number, 28 Spectrum of modules, 409 Transformation group, 116, 138 Spectrum of rings, 410 Transition matrix, 304 Splitting field, 498 Transitive, 8 Splitting homomorphism, 388 Transitive action, 139 Square and multiply algorithm, 461 Trapdoor of a function, 529 Stabilizer, 139 Triangular matrix, 344 State machine, 155 Trisection of an angle, 513 Statement, 2 Trivial ideals, 205 Stereographic projection, 24 Twin primes, 419 Strong mathematical induction, 47 Structure space, 597 U Structure theorem for groups, 383 Uniform distribution, 480 Structure theorem for modules, 381 Union, 4 Subalgebra, 300 Union of sets, 4 Subfield, 168 Unique factorization domain, 242 Subgroup, 83 Unit, 163 Subgroupoid, 57 Unit vector, 327 Submodule, 358 Unitary matrix, 314 Subring, 166 Unitary module, 356 Subspace, 277 Universal set, 4 Subspace generated, 281 Urysohn lemma, 223 Substitution cipher, 523 Substitution homomorphism, 183 V Sum of divisor function, 423 Vandermonde, 621 Sum of ideals, 205 Vector, 275 Sum of subspaces, 278 Vector space, 274 Sum of the submodules, 363 Venn diagrams, 5 Surjective, 21 Vigenère cipher, 525 Sylow, 626 Sylow’s theorem, 142 W Sylvester’s law of nullity, 295 Wedderburn, 629 Sylvester’s theorem, 337 Wedderburn Artin theorem, 405 Symmetric, 8 Wedderburn theorem, 189 Symmetric difference, 5 Well defined, 2 Symmetric group, 75 Well-ordering principle, 41 Symmetric key, 527 Weyl, 621 Symmetry, 99 Wiles, vii Wilson’s theorem, 426 T Tensor product, 404 Y The Eisenstein criterion, 247 Yes-biased Monte Carlo algorithm, 466 Theorem of quotient of quotient of a sum for Yoneda’s lemma, 610 modules, 372 Third isomorphism theorem for modules, 372 Z Third Sylow theorem, 148 Zariski, 629 Three lemma, 389 Zariski topology, 218 Threshold scheme, 548 Zero divisor, 161 Topological group, 112, 151 Zero ring, 161 Topological rings, 197 Zorn’s lemma, 16