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Revision of Ring Theory CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic object in which we can operate in a similar way as we do with integers. Inside the set Z of integers we can perform operations of addition, substraction and multiplication, but in general Z is not closed under division; for instance the quotient 3/2 is not an integer. Addition and multiplication have a series of well known properties; the abstraction of those properties is what constitutes the formal notion of ring. DEFINITION 1.1.1. A ring is a nonempty set R together with two operations, a sum + and a product satisfying the following properties: · Sum S1 Associativity: (a + b)+c = a +(b + c) for all a, b, c R, ∈ S2 Commutativity: a + b = b + a for all a, b R, ∈ S3 Zero: There is an element 0 R such that a +0=a =0+a for each a R, ∈ ∈ S4 Inverses: For each a R there is an element a R such that a +( a)=0. ∈ − ∈ − In what follows we will always write a b to denote a +( b). − − REMARK. These properties simply mean that (R, +) is an abelian group. Product P1 Associativity: a(bc)=(ab)c for all a, b, c R, ∈ P2 Unit: There is an element 1 R such that 1a = a1=a for all a R, ∈ ∈ REMARK. This properties mean that (R, ) is a multiplicative monoid. · P3 Distributivity: For all a, b, c R, ∈ a(b + c)=ab + ac, (a + b)c = ac + bc. Some rings satisfy an additional property for the product: P4 Commutativity: ab = ba for all a, b R. ∈ When this extra property is satisfied, we will say that R is a commutative ring. In this course, we will be mostly interested on commutative rings, although we will occasionally deal with some noncommutative examples. EXAMPLE 1.1.2. R = 0 with the trivial operations. This is called the trivial ring, { } and is the only ring for which one has 1=0. In all what follows, we will assume our rings to be nontrivial, i.e. 0 =1. EXAMPLE 1.1.3. The integers Z with the usual addition and multiplication. EXAMPLE 1.1.4. The fields of rational numbers Q, real numbers R or complex num- bers C, or in general any field F. 1 2 1. REVISION OF RING THEORY EXAMPLE 1.1.5. The rings of integers modulo n, Z/nZ (sometimes also denoted by Zn) consisting of the set a a Z = 0, 1,...,n 1 , where a is the residue class { | ∈ } { − } of a modulo n, so a = a = rn r Z . Addition and multiplication are defined as { | ∈ } a + b := a + b, ab := ab EXAMPLE 1.1.6. Let R be any ring, and define R[x] to be the set of all polynomials a + a x + + a xn where the coefficients a are elements of the ring R. Then R[x] 0 1 ··· n i is a ring with addition and multiplication defined in the usual way (check as an exercise!). The ring R[x] is called the polynomial ring in one variable with coefficients in R. EXAMPLE 1.1.7. The polynomial ring in n variables with coefficients in R, denoted by R[x1,...,xn] and defined inductively by R[x1,...,xn]:=R[x1,...,xn 1][xn]. − EXAMPLE 1.1.8. The ring M (R) of n n matrices (a ) with coefficients n × i,j i,j=1,...,n ai,j in R, and the usual matrix addition and multiplication. EXAMPLE 1.1.9. The power set ring. Let X be a set, and let (X)= Y Y X P { | ⊆ } the set of all subsets of X. On (X) consider the operations: P Y + Z := Y Z =(Y Z) (Y Z) the symmetric difference as addition, • ∪ \ ∩ YZ:= Y Z the intersection as a product. • ∩ With this operations (X) becomes a ring with zero element 0=∅ and unit element P 1=X. EXAMPLE 1.1.10. Let V be a vector space, consider the set End(V ):= f : V { → V f is a linear map , then End(V ) is a ring with pointwise addition (f +g)(v):=f(v)+ | } g(v) and multiplication given by composition fg(v):=f(g(v)), where the zero element is the constant map 0(v)=0and the unit element is the identity map Id(v)=v. REMARK. The ring of endomorphisms of a vector space is nothing but a matrix ring in disguise. We will state more precisely what we mean by this once we talk about ring isomorphisms. EXAMPLE 1.1.11. Consider the set C(R):= f : R R f continuous function of { → | } real-valued continuous functions. The C(R) is a ring with pointwise addition (f +g)(x):= f(x)+g(x) and multiplication (fg)(x):=f(x)+g(x). REMARK. One might wonder whether one could define a different ring structure on C(R) by replacing pointwise multiplication by composition as a product. Unlike it hap- pened in the case of linear map, this operation does not turn C(R) into a ring. EXAMPLE 1.1.12. Let X be a set and R be a ring. In a similar fashion to the previous example, the set XR of all R-valued maps f : X R on X becomes naturally a ring with → pointwise addition and multiplication. EXAMPLE 1.1.13. Quaternion algebras. Let F be a field (of characteristic different from 2), and let α,β F. The quaternion algebra αFβ is defined as the set a + bi + ∈ { cj + dk a, b, c, d F with standard sum and product defined by the rules ij = k = ji, | ∈ } − i2 = α, j2 = β. If F is a subfield of the real numbers, αFβ can also be described as the subring a + b√αc√β + d√αβ of M ( ) consisting of matrices of the form , where 2 C c√β d√αβ a b√α − − a, b, c, d F. ∈ EXAMPLE 1.1.14. The ring of power series with real coefficients n R[[x]] := anx an R n . | ∈ ∀ n 0 ≥ In general one can construct the ring of formal power series R[[x]] with coefficients in any ring R. 1.2. SUBRINGS, IDEALS AND QUOTIENT RINGS 3 EXAMPLE 1.1.15. Let G be a group, R a ring, the group ring R[G] is defined as R[G]:= a x a R x, a =0for all x except a finite number x G x | x ∈ ∀ x = f : G∈ R f has a finite support . { → | } The addition and product are defined (in the functional notation) as follows: (f + g)(x):=f(x)+g(x), • 1 (fg)(x):=(f g)(x)= y G f(y)g(y− x). • ∗ ∈ The product in R[G] receives the name of convolution product of functions. REMARK. Note that in this case the convolution product actually defines a different ring structure in the set of functions f : G R than the pointwise product, so this example → is actually different from Example 1.1.12; for instance, if R is a commutative ring then GR (with the pointwise product) is also commutative, whereas the group ring R[G] will be noncommutative whenever G is. 1.2. Subrings, ideals and quotient rings Let R be a ring, we look at subsets of R which are in fact themselves rings in their own right when we restrict to them the sum and product of R. More precisely, we say that S is a subring of R if: (i) 1 S, R ∈ (ii) S is an additive subgroup of R. In other words, whenever a, b S, one has a b S. ∈ − ∈ (iii) S is closed under the product of R, in other words, S is a multiplicative submonoid of R, i.e. for all a, b S one has ab S. ∈ ∈ We will write S R to denote that S is a subring of R. ≤ EXAMPLES 1.2.1. (1) Z Q R C. ≤ ≤ ≤ (2) The trivial ring 0 is NOT a subring of Z, as 1 / 0 . { } ∈{ } (3) For any ring R, one has R R[x], where R consists of all constant polynomials ≤ in R[x]. (4) The rings of matrices Mn(R) contain several interesting subrings. Some exam- ples are The ring D (R) of diagonal matrices. • n The ring U (R) of upper-triangular matrices. • n The ring L (R) of lower-triangular matrices. • n (5) If Si i I is a family of rings such that Si R for all i I, then Si R. { } ∈ ≤ ∈ i I ≤ (6) Let X R be a subset of a ring R, define [X]:= S R X∈ S the ⊆ { ≤ | ⊆ } intersection of all subrings of R containing X. Then [X] is a subring of R, called the subring generated by X. EXERCISE 1.2.2. Show that [X] can be identified with the set of all sums of the form x x where x X 1 . ± 1 ··· n i ∈ ∪{ } We move now to the key notion of ideal. Ideals are certain subsets of rings that play a similar role to that of normal subgroups in group theory, in the sense they allow us to build quotients of rings. Also, knowing the ideals of a ring in full detail often lead to a complete description of all the modules, so understanding ideals is a fundamental topic of this course. DEFINITION 1.2.3. Let R be a commutative ring. A subset I R of R is said to be ⊆ an ideal of R if it has the following properties: I1 Additive closure: I (R, +) is an additive subgroup of R, i.e.
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