DOCSLIB.ORG

Zero Divisor Conjecture for Group Semifields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Zero Divisor Conjecture for Group Semifields Ultra Scientist Vol. 27(3)B, 209-213 (2015). Zero Divisor Conjecture for Group Semifields W.B. VASANTHA KANDASAMY1 1Department of Mathematics, Indian Institute of Technology, Chennai- 600 036 (India) E-mail: [email protected] (Acceptance Date 16th December, 2015) Abstract In this paper the author for the first time has studied the zero divisor conjecture for the group semifields; that is groups over semirings. It is proved that whatever group is taken the group semifield has no zero divisors that is the group semifield is a semifield or a semidivision ring. However the group semiring can have only idempotents and no units. This is in contrast with the group rings for group rings have units and if the group ring has idempotents then it has zero divisors. Key words: Group semirings, group rings semifield, semidivision ring. 1. Introduction study. Here just we recall the zero divisor For notions of group semirings refer8-11. conjecture for group rings and the conditions For more about group rings refer 1-6. under which group rings have zero divisors. Throughout this paper the semirings used are Here rings are assumed to be only fields and semifields. not any ring with zero divisors. It is well known1,2,3 the group ring KG has zero divisors if G is a 2. Zero divisors in group semifields : finite group. Further if G is a torsion free non abelian group and K any field, it is not known In this section zero divisors in group whether KG has zero divisors. In view of the semifields SG of any group G over the 1 problem 28 of . However in this paper it is semifields (Z+ {0}, R+ {0}, Q+ {0}, Z proved whatever be the group G, the group semiring SG where S is a semifield has no zero I {0}, R I {0}, Q I divisors. This paper has three sections. Section one is introductory in nature. Section two {0} and distributive lattices L such that ai studies zero divisors in group semifields and aj 0 for ai, aj L \ {0}) are studied. It is section three gives conclusions based on our proved SG is a semifield if G is commutative 210 W.B. Vasantha Kandasamy and SG is a semidivision ring if G is non- First we will illustrate this situation by some commutative. examples. For more about semifields refer11. Just for the sake of completeness of Example 2.1. Let S =R+ {0} be the this paper the definition of group semifield is 24 recalled. semifield of reals. G = g g 1 be the cyclic group of order 24. SG be the group Definition 2.1: Let S be a semifield semifield of finite order. and G any group. SG = {Collection of all SG has no zero divisors. SG has no finite formal sums of the form si g i , i runs i units other than x = 1.gi where gi G; which over a finite index; si S and gi G} is is known as the trivial units. SG has idempo- defined as the group semifield of the group tents. G over the semifield S if the following 1 conditions are satisfied. For take = (1 + g12) SG.. 2 n n 2 ig i i g i 1 1 (i) if and only if i =i 2 12 12 24 i1 i 1 = (1 g ) = (1 + 2g + g ) 2 4 for i = 1, 2, …, n. n n n 1 = (2 + 2g12) (ii) ig i i g i () i i g i 4 i1 i 1 i 1 1 12 n n t = (1 + g ) 2 (iii) ig i i g i k m k wheree i1 i 1 k 1 = . n However in this group semiring the i j k = ; mk = gihj. presence of this idempotent does not lead to i 1 zero divisors as in case of rings. (iv) ri gi = gi ri for all ri S and gi G. n n 1 = (1 + g6 + g12 + g18) is again an (v) s si g i ss i g i for r S and 4 i1 i 1 idempotent of SG. n 1 si g i SG.. = (1 + g8 + g16) SG is an i 1 3 idempotent of SG. SG is a semiring/semifield with 0 as the additive identity. As 1 S and 1 G. 1 = (1 + g3 + g6 + g9 + g12 + g15 + 1.G SG and S.1 SG. 8 Zero Divisor Conjecture for Group Semifields. 211 g18 + g21) SG is an idempotent. of them. The distributive lattices with the Hasse diagram which are semifields is as 1 y = (1 + g2 + g4 + … + g22) SG follows: 12 1 is an idempotent of SG. a1 1 4 8 12 16 20 a2 = (1 + g + g + g + g + g ) 6 a3 L1= is an idempotent of SG. a 4 a However SG is only a semifield as SG 5 a is commutative. 6 0 Note: In the above example if the real + 1 semifield R {0} is replaced by the rational semifield Q+ {0}, the above conclusions are a true. a a But however if the real semifield L R+ {0} is replaced by the integer semifield a Z+ {0} then the group semifield has no idempotents. Interested reader can replace a R+ {0} by R I {0}, Q I {0} a6 a 7 and Z I {0} and study this problem; it 0 is left as a matter of routine. However Z Consider, 2 1 I {0} has no idempotents other than I = I. a Next one example in which a a distributive lattice of finite order which is a a semifield is used. We say a distributive lattice L with 0 and 1. Lattice L is said to have a zero a divisor if ai aj = 0 where ai, aj L \ {0}. Such distributive lattices are declared as a5 a6 semifields which has no zero divisors. So in this paper, finite a distributive lattice L with 0 a7 and 1 will be termed as semifields of finite order. First we will give one or two examples 0 212 W.B. Vasantha Kandasamy 1 symmetric group of degree seven. L = C10 be the chain lattice 0 < a8 < a7 < … < a1 < 1. LG be the group semifield. LG is a non-commutative L a a semiring. LG has no zero divisors and units. However LG has idempotents. LG is a 0 semidivision ring of finite order. 1 In view of all these the following observations are made. a L a a The group semifields SG of all groups G over every semifield S (finite semifields or a a otherwise) has no zero divisors only trivial units a and idempotents which are nontrivial. 0 However if S = Z+ {0}, the semifield of integers; SG the group semifield where G and is any group. SG has no idempotents, non trivial 1 units and zero divisors. Theorem 2.1: Let G be any group a a finite or infinite commutative or non- commutative. S be any semifield, SG the a group semifield. SG has no zero divisors 3 and SG is a semifield. L a4 Proof: Follows from the fact S has no zero divisors and S is a strict semiring so a a5 6 SG cannot have zero divisors as . = 0 is impossible for any , SG \ {0}. 0 Corollary: If in the above theorem S These three distributive lattices are is taken as a semiring with zero divisors. For only semirings as they contain zero divisors. every group G, SG, the group semiring has zero divisors. Those group semiring using the semirings Proof is direct. L4 or L5 or L6 will be a semiring with zero divisors. Next a brief comparison is made in Example 2.2: Let G = S7 be the the following section. Ultra Scientist Vol. 27(3)B, (2015). 213 3. Conclusions Acad., Vol. 60, 333-334, (1984). 4. Vasantha Kandasamy W.B., On semi It is proved SG the group semifield of idempotents in group rings, Proc. of the any group G over any semifield S is a semifield Japan Acad., Vol. 61, 107-108, (1985). or a semidivision ring. Secondly when SG the 5. Vasantha Kandasamy W.B., A note on group semifield has idempotents still SG has the modular group ring of finite p-group, no zero divisors. This is in sharp contrast Kyungpook Math. J., Vol. 25, 163-166, between the group rings RG where R is a field (1986). and G any group has zero divisors if G is a 6. Vasantha Kandasamy W.B., Idempotents finite group or a torsion group. Thus the famous in the group ring of a cyclic group, Vikram open conjecture for group rings has no Math. Journal, Vol. X, 59-73, (1990). relevance in case of group semifields. 7. Vasantha Kandasamy W.B., Semi group rings of ordered semigroups which are Finally another big contrast is existence reduced rings, J. of Math. Res. and of idempotents does not lead to zero divisors Expo., Vol. 10, 494-493, (1990). in group semifields however in group rings the 8. Vasantha Kandasamy W.B., Semigroup presence of idempotents guarantees the rings which are p-rings, Bull. Calcutta presence of zero divisors. Math. Soc., Vol. 82, 191-192, (1990). 9. Vasantha Kandasamy W.B., A note on References semigroup rings which are Boolean rings, Ultra Sci. of Phys. Sci., Vol. 3, 1. Passman, D.S., Infinite Group Rings, 67-68, (1991). Pure and Appl. Math., Marcel Dekker, 10. Vasantha Kandasamy W.B., On semigroup (1971). rings which are Marot rings, Revista 2. Passman, D.S., The Algebraic Structure Integracion, Vol.
Load more

Recommended publications
  • Adjacency Matrices of Zero-Divisor Graphs of Integers Modulo N Matthew Young
    inv lve a journal of mathematics Adjacency matrices of zero-divisor graphs of integers modulo n Matthew Young msp 2015 vol. 8, no. 5 INVOLVE 8:5 (2015) msp dx.doi.org/10.2140/involve.2015.8.753 Adjacency matrices of zero-divisor graphs of integers modulo n Matthew Young (Communicated by Kenneth S. Berenhaut) We study adjacency matrices of zero-divisor graphs of Zn for various n. We find their determinant and rank for all n, develop a method for finding nonzero eigen- values, and use it to find all eigenvalues for the case n p3, where p is a prime D number. We also find upper and lower bounds for the largest eigenvalue for all n. 1. Introduction Let R be a commutative ring with a unity. The notion of a zero-divisor graph of R was pioneered by Beck[1988]. It was later modified by Anderson and Livingston [1999] to be the following. Definition 1.1. The zero-divisor graph .R/ of the ring R is a graph with the set of vertices V .R/ being the set of zero-divisors of R and edges connecting two vertices x; y R if and only if x y 0. 2 D To each (finite) graph , one can associate the adjacency matrix A./ that is a square V ./ V ./ matrix with entries aij 1, if vi is connected with vj , and j j j j D zero otherwise. In this paper we study the adjacency matrices of zero-divisor graphs n .Zn/ of rings Zn of integers modulo n, where n is not prime.
    [Show full text]
  • On Finite Semifields with a Weak Nucleus and a Designed
    On finite semifields with a weak nucleus and a designed automorphism group A. Pi~nera-Nicol´as∗ I.F. R´uay Abstract Finite nonassociative division algebras S (i.e., finite semifields) with a weak nucleus N ⊆ S as defined by D.E. Knuth in [10] (i.e., satisfying the conditions (ab)c − a(bc) = (ac)b − a(cb) = c(ab) − (ca)b = 0, for all a; b 2 N; c 2 S) and a prefixed automorphism group are considered. In particular, a construction of semifields of order 64 with weak nucleus F4 and a cyclic automorphism group C5 is introduced. This construction is extended to obtain sporadic finite semifields of orders 256 and 512. Keywords: Finite Semifield; Projective planes; Automorphism Group AMS classification: 12K10, 51E35 1 Introduction A finite semifield S is a finite nonassociative division algebra. These rings play an important role in the context of finite geometries since they coordinatize projective semifield planes [7]. They also have applications on coding theory [4, 8, 6], combinatorics and graph theory [13]. During the last few years, computational efforts in order to clasify some of these objects have been made. For instance, the classification of semifields with 64 elements is completely known, [16], as well as those with 243 elements, [17]. However, many questions are open yet: there is not a classification of semifields with 128 or 256 elements, despite of the powerful computers available nowadays. The knowledge of the structure of these concrete semifields can inspire new general constructions, as suggested in [9]. In this sense, the work of Lavrauw and Sheekey [11] gives examples of constructions of semifields which are neither twisted fields nor two-dimensional over a nucleus starting from the classification of semifields with 64 elements.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • Rings of Quotients and Localization a Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Ar
    RINGS OF QUOTIENTS AND LOCALIZATION A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREME N TS FOR THE DEGREE OF MASTER OF ARTS IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE TEXAS WOMAN'S UNIVERSITY COLLEGE OF ARTS AND SCIENCE S BY LILLIAN WANDA WRIGHT, B. A. DENTON, TEXAS MAY, l974 TABLE OF CONTENTS INTRODUCTION . J Chapter I. PRIME IDEALS AND MULTIPLICATIVE SETS 4 II. RINGS OF QUOTIENTS . e III. CLASSICAL RINGS OF QUOTIENTS •• 25 IV. PROPERTIES PRESERVED UNDER LOCALIZATION 30 BIBLIOGRAPHY • • • • • • • • • • • • • • • • ••••••• 36 iii INTRODUCTION The concept of a ring of quotients was apparently first introduced in 192 7 by a German m a thematician Heinrich Grell i n his paper "Bezeihungen zwischen !deale verschievener Ringe" [ 7 ] . I n his work Grelt observed that it is possible to associate a ring of quotients with the set S of non -zero divisors in a r ing. The elements of this ring of quotients a r e fractions whose denominators b elong to Sand whose numerators belong to the commutative ring. Grell's ring of quotients is now called the classical ring of quotients. 1 GrelL 1 s concept of a ring of quotients remained virtually unchanged until 1944 when the Frenchman Claude C hevalley presented his paper, "On the notion oi the Ring of Quotients of a Prime Ideal" [ 5 J. C hevalley extended Gre ll's notion to the case wher e Sis the compte- ment of a primt: ideal. (Note that the set of all non - zer o divisors and the set- theoretic c om?lement of a prime ideal are both instances of .multiplicative sets -- sets tha t are closed under multiplicatio n.) 1According to V.
    [Show full text]
  • Mathematics 411 9 December 2005 Final Exam Preview
    Mathematics 411 9 December 2005 Final exam preview Instructions: As always in this course, clarity of exposition is as important as correctness of mathematics. 1. Recall that a Gaussian integer is a complex number of the form a + bi where a and b are integers. The Gaussian integers form a ring Z[i], in which the number 3 has the following special property: Given any two Gaussian integers r and s, if 3 divides the product rs, then 3 divides either r or s (or both). Use this fact to prove the following theorem: For any integer n ≥ 1, given n Gaussian integers r1, r2,..., rn, if 3 divides the product r1 ···rn, then 3 divides at least one of the factors ri. 2. (a) Use the Euclidean algorithm to find an integer solution to the equation 7x + 37y = 1. (b) Explain how to use the solution of part (a) to find a solution to the congruence 7x ≡ 1 (mod 37). (c) Use part (b) to explain why [7] is a unit in the ring Z37. 3. Given a Gaussian integer t that is neither zero nor a unit in Z[i], a factorization t = rs in Z[i] is nontrivial if neither r nor s is a unit. (a) List the units in Z[i], indicating for each one what its multiplicative inverse is. No proof is necessary. (b) Provide a nontrivial factorization of 53 in Z[i]. Explain why your factorization is nontriv- ial. (c) Suppose that p is a prime number in Z that has no nontrivial factorizations in Z[i].
    [Show full text]
  • Chapter 10 an Introduction to Rings
    Chapter 10 An Introduction to Rings 10.1 Definitions and Examples Recall that a group is a set together with a single binary operation, which together satisfy a few modest properties. Loosely speaking, a ring is a set together with two binary oper- ations (called addition and multiplication) that are related via a distributive property. Definition 10.1. A ring R is a set together with two binary operations + and (called addition and multiplication, respectively) satisfying the following: · (i) (R,+) is an abelian group. (ii) is associative: (a b) c = a (b c) for all a,b,c R. · · · · · 2 (iii) The distributive property holds: a (b + c)=(a b)+(a c) and (a + b) c =(a c)+(b c) · · · · · · for all a,b,c R. 2 Remark 10.2. We make a couple comments about notation. (a) We often write ab in place a b. · (b) The additive inverse of the ring element a R is denoted a. 2 − Theorem 10.3. Let R be a ring. Then for all a,b R: 2 1. 0a = a0=0 2. ( a)b = a( b)= (ab) − − − 3. ( a)( b)=ab − − Definition 10.4. A ring R is called commutative if multiplication is commutative. Definition 10.5. A ring R is said to have an identity (or called a ring with 1) if there is an element 1 R such that 1a = a1=a for all a R. 2 2 Exercise 10.6. Justify that Z is a commutative ring with 1 under the usual operations of addition and multiplication. Which elements have multiplicative inverses in Z? CHAPTER 10.
    [Show full text]
  • Student Notes
    270 Chapter 5 Rings, Integral Domains, and Fields 270 Chapter 5 Rings, Integral Domains, and Fields 5.2 Integral5.2 DomainsIntegral Domains and Fields and Fields In the precedingIn thesection preceding we definedsection we the defined terms the ring terms with ring unity, with unity, commutative commutative ring, ring,andandzerozero 270 Chapter 5 Rings, Integral Domains, and Fields 5.2divisors. IntegralAll Domains threedivisors. of and theseAll Fields three terms ! of these are usedterms inare defining used in defining an integral an integral domain. domain. Week 13 Definition 5.14DefinitionI Integral 5.14 IDomainIntegral Domain 5.2 LetIntegralD be a ring. Domains Then D is an andintegral Fields domain provided these conditions hold: Let D be a ring. Then D is an integral domain provided these conditions hold: In1. theD ispreceding a commutative section ring. we defined the terms ring with unity, commutative ring, and zero 1. D is a commutativedivisors.2. D hasAll a ring. unity three eof, and thesee 2terms0. are used in defining an integral domain. 2. D has a unity3. De,has and noe zero2 0 divisors.. Definition 5.14 I Integral Domain 3. D has no zero divisors. Note that the requirement e 2 0 means that an integral domain must have at least two Let D be a ring. Then D is an integral domain provided these conditions hold: Remark. elements. Note that the1. requirementD is a commutative e 2 0ring. means that an integral domain must have at least two elements. Example2. D has a1 unityThe e, ring andZe of2 all0. integers is an integral domain, but the ring E of all even in- tegers3.
    [Show full text]
  • FUSIBLE RINGS 1. Introduction It Is Well-Known That the Sum of Two Zero
    FUSIBLE RINGS EBRAHIM GHASHGHAEI AND WARREN WM. MCGOVERN∗ Abstract. A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x; σ] and R[[x; σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semi-simple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full Rings of continuous functions are fusible. It is also shown that if 1 = e1 + e2 + ::: + en in a ring R where the ei are orthogonal idempotents and each eiRei is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings. Keywords: Fusible rings, unit fusible rings, zero-divisors, clean rings, C(X). MSC(2010): Primary:16U99; Secondary:16W99, 13A99. 1. Introduction It is well-known that the sum of two zero-divisors need not be a zero-divisor. In [18, Theorem 1.12] the authors characterized commutative rings for which the set of zero-divisors is an ideal. The set of zero-divisors (i.e., Z(R)) is an ideal in a commutative ring R if and only if the classical ring of quotients of R (i.e., qcl(R)) is a local ring if and only if Z(R) is a prime ideal of R and qcl(R) is the localization at Z(R).
    [Show full text]
  • MODEL ANSWERS to HWK #7 1. Suppose That F Is a Field and That A
    MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F . Suppose that a · b = 0; and that b 6= 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0 = 0 · c = (a · b) · c = a · (b · c) = a · 1 = a: Thus a = 0. It follows that F is an integral domain. 2. We compute a b 1 0 a 0 = c d 0 0 c 0 and 1 0 a b a b = : 0 0 c d 0 0 If we have equality, then b = 0 and c = 0 and vice-versa. Thus the only matrices that commute with 1 0 0 0 are matrices of the form a 0 ; 0 b where a and b are arbitrary. 3. (a) One way to do this is to write down three general matrices and compute the triple product explicitly, both ways, and check that you get the same answer. A simpler, and possibly more aesthetically pleasing approach, is as follows. Note that a 2×2 matrix A determines a map f : R2 −! R2; where R2 = R×R is the ordinary cartesian product. Given v = (a; b) 2 R2, formally treat this as a column vector and define f(a; b) = Av; 1 where Av is computed in the usual way. Note that, in these terms, the product of two matrices corresponds to composition of functions, and that f determines A; indeed the top row of the matrix can be recovered from Av, where v = (1; 0) and the bottom row from Av, where v = (0; 1).
    [Show full text]
  • Constructing Zero Divisor Graphs
    Introduction/Background Our Methods Results and Conclusions References Constructing Zero Divisor Graphs Alaina Wickboldt, Louisiana State University Alonza Terry, Xavier University of Louisiana Carlos Lopez, Mississippi State University SMILE 2011 A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References Outline Introduction/Background Ring Theory Modular Arithmetic Zero Divisor Graphs Our Methods Drawing by hand Implementing Computer Coding Results and Conclusions Results Conclusion References A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References Joint work with: • Dr. Sandra Spiroff, University of Mississippi • Dr. Dave Chapman, Louisiana State University A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs • Construct the [AL] graph Γ(R) for R = (Zn) up to order n, for n ≤ 100. • Observe behavior and patterns of each graph. Introduction/Background Our Methods Results and Conclusions References THE PROJECT To study the various definitions of zero divisor graphs associated to rings, namely those by I. Beck [B] and D. Anderson & P. Livingston [AL]. A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References THE PROJECT To study the various definitions of zero divisor graphs associated to rings, namely those by I. Beck [B] and D. Anderson & P. Livingston [AL]. • Construct the [AL] graph Γ(R) for R = (Zn) up to order n, for n ≤ 100. • Observe behavior and patterns of each graph. A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Ring Theory Our Methods Modular Arithmetic Results and Conclusions Zero Divisor Graphs References Relations Between Rings and Graphs Ring Graph 2 1 < ¢¢ << ¢¢ ¢¢ << ¢¢ ¢¢ << ¢¢ ¢¢ < ¢¢ 3 0 G( ) < < Z6 << ¢¢ << << ¢¢ << Z/6Z << ¢¢ << < ¢¢ < 4 5 A.
    [Show full text]
  • 28. Com. Rings: Units and Zero-Divisors
    M7210 Lecture 28 Friday October 26, 2012 Commutative Rings III We assume all rings are commutative and have multiplicative identity (different from 0). Units and zero-divisors Definition. Let A be a commutative ring and let a ∈ A. i) a is called a unit if a 6=0A and there is b ∈ A such that ab =1A. ii) a is called a zero-divisor if a 6= 0A and there is non-zero b ∈ A such that ab = 0A. A (commutative) ring with no zero-divisors is called an integral domain. Remark. In the zero ring Z = {0}, 0Z = 1Z. This is an annoying detail that our definition must take into account, and this is why the stipulation “a 6= 0A” appears in i). The zero ring plays a very minor role in commutative algebra, but it is important nonetheless because it is a valid model of the equational theory of rings-with-identity and it is the final object in the category of rings. We exclude this ring in the discussion below (and occasionally include a reminder that we are doing so). Fact. In any (non-zero commutative) ring R the set of zero-divisors and the set of units are disjoint. Proof . Let u ∈ R \{0}. If there is z ∈ R such that uz = 0, then for any t ∈ R, (tu)z = 0. Thus, there is no t ∈ R such that tu = 1. Fact. Let R be a ring. The set U(R) of units of R forms a group. Examples. The units in Z are 1 and −1.
    [Show full text]
  • Math 403 Chapter 13: Integral Domains and Fields 1. Introduction: Rings Are Closer to Familiar Structures Like R in That They Ge
    Math 403 Chapter 13: Integral Domains and Fields 1. Introduction: Rings are closer to familiar structures like R in that they get addition (therefore subtraction) and multiplication, but they don't necessarily get division. More generally certain standard assumptions break down. For example in R we know that if ab = 0 then a = 0 or b = 0 but this isn't necessarily the case in a general ring. What we'll do in this chapter is focus on certain types of ring in which behavior is a little more familiar. 2. Integral Domains: (a) Definition: If R is a commutative ring then a 2 R is a zero-divisor if there is some b 2 R with ab = 0. Example: In Z10 the integer 5 is a zero-divisor because (5)(2) = 0 whereas 3 is not a zero-divisor because there is no b 2 Z10 with 3b = 0. Note: When defining a zero-divisor we requite R to be commutative to avoid issues that arise if ab = 0 but ba 6= 0. This can happen in rings, for example in M2R: 1 0 0 0 0 0 0 0 1 0 0 0 = but = 0 0 1 0 0 0 1 0 0 0 1 0 This complicates the definition as to whether we should consider a and b to be zero- divisors. (b) Definition: A commutative ring with a unity is an integral domain if it has no zero- divisors. In other words a commutative ring with unity is an integral domain if, whenever ab = 0, we must have a = 0 or b = 0.
    [Show full text]