Chapter 2 Rings and Modules 2.1 Rings Definition 2.1.1. A ring consists of a set R together with binary operations + and · satisfying: 1. (R; +) forms an abelian group, 2. (a · b) · c = a · (b · c) 8a; b; c 2 R, 3. 91 =6 0 2 R such that a · 1 = 1 · a = a 8a 2 R, and 4. a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c 8a; b; c 2 R. Note: 1. Some people (e.g. Dummit + Foote) do not require condition 3, and refer to a \ring with identity" if they want to assume · has an identity element. 2. People who include existence of a unit in their defn. of a ring refer to a \ring without identity" for an object satisfying the other three axioms. Some people (e.g. Jacobson) call this a \rng". 3. Some people (e.g. Lang) do not require 1 =6 0 in condition 3. Definition 2.1.2. R is called commutative if its multiplication is commutative, ie. ab = ba 8a; b 2 R: Definition 2.1.3. A ring homomorphism from R to S is a function f : R 7! S such that 8a; b 2 R: 55 1. f(a + b) = f(a) + f(b), 2. f(ab) = f(a)f(b), and 3. f(1) = 1. A bijective ring homomorphism is called an isomorphism. Definition 2.1.4. A subring of R is a subset A which forms a ring such that the inclusion A ,! R is a ring homomorphism. A subgroup I of the abelian group (R; +) is called a (two - sided) ideal if x 2 I; r 2 R ) rx 2 I and xr 2 I: Similarly if a subgroup I satisfies x 2 I; r 2 R ) rx 2 I; I is called a left ideal, and if it satisfies x 2 I; r 2 R ) xr 2 I; it is called a right ideal. Example 2.1.5. If f : R 7! S is a homomorphism then ker f := fx 2 R j f(x) = 0g is an ideal in R. (An ideal is always a subrng but never a subring, unless it is all of R.) Theorem 2.1.6. Let I⊆R be a proper ideal. Then 9 a ring R=I and a surjective ring homo- morphism f : R 7!! R=I such that ker f = I. Proof. Define an equivalence relation on R by x ∼ y () x − y 2 I. Let R=I := fequiv. classesg: Define operations on R=I by [x] + [y] := [x + y]; [x] · [y] := [xy]: Check that these are well-defined and produce a ring structure on R=I. Define f : R 7! R=I by f(x) = [x]. f is a ring homomorphism. Moreover, f(x) = 0 iff [x] = 0 iff x = x − 0 2 I. 56 Definition 2.1.7. The ring R is called a division ring if (R − f0g; ·) forms a group. A commutative division ring is called a field. An element u 2 R for which 9v 2 R such that uv = vu = 1 is called a unit. Notation: R× = funits of Rg: This forms a group under multiplication. A non-zero element x 2 R is called a zero divisor if 9y =6 0 such that either xy = 0 or yx = 0. A commutative ring with no zero divisors is called an integral domain. Proposition 2.1.8. If x =6 0 is not a zero divisor and xy = xz then y = z. Proof. x(y − z) = 0 and x is not a zero divisor so either x = 0 or y − z = 0. But x =6 0 so y = z. Theorem 2.1.9 (First Isomorphism Theorem). Let f : R 7! S be a ring homomorphism. Then R= ker f =∼ Imf. Theorem 2.1.10 (Second Isomorphism Theorem). Let A ⊂ R be a subring and let I⊆R be a proper ideal. Then A + I := fa + x j a 2 A; x 2 Ig is a subring of R, A \ I is a proper ideal in A, and (A + I)=I ∼= A=(A \ I): Theorem 2.1.11 (Third Isomorphism Theorem). Let I ⊂ J be proper ideals of R. Then J=I := f[x] 2 R=I j x 2 Jg is an ideal in R=I, and R=I ∼= R=J: J=I Theorem 2.1.12 (Fourth Isomorphism Theorem). Let I be a proper ideal of R. Then the correspondence J 7! J=I is a bijection between the ideals of J containing I and the ideals of R=I. Let I; J be ideals in R. Define ideals I + J := fx + y j x 2 I; y 2 Jg; I \ J; n IJ := xiyy j n 2 N; xi 2 I; yi 2 J ( i=1 ) X Then IJ ⊂ I \ J ⊂ I [ J ⊂ I + J: (Note that I [ J may not be an ideal.) I + J is the smallest ideal containing both I and J. 57 2.2 Maximal and Prime Ideals Definition 2.2.1. An ideal M⊆R is called a maximal ideal if @ an ideal I s.t. M⊆I⊆R. Lemma 2.2.2. Given an ideal I⊆R, 9 a maximal ideal M s.t. I ⊂ M. Proof. Let S = fideals J j I ⊂ J⊆Rg: Then S is a partially ordered set (ordered by inclusion). If C ⊂ S is a chain (ie. a totally ordered subset) then J = C C [2C is an ideal which forms an upper bound for C in S (it is indeed a proper ideal since 1 62 J). ) Zorn's Lemma ) S has a maximal element M. For the rest of this section, suppose that R is commutative. Proposition 2.2.3. R is a field () the only ideals of R are f0g and R. Proof. ): Let R be a field and let I ⊂ R be an ideal. If I =6 f0g then 9x =6 0 2 I. R a field ) 9y 2 R such that xy = yx = 1. Since I is an ideal, 1 2 I, so r 2 I 8r 2 R. Thus I = R. (: Suppose the only ideals in R are f0g and R. Let x =6 0 2 R. Let I = Rx := frx j r 2 Rg: I is an ideal and x = 1x 2 R, so I =6 0. Hence I = R, so 1 2 I. ie. 1 = yz for some y 2 R. ) Every x =6 0 2 R has an inverse, so R is a field. Corollary 2.2.4. Let f : F 7! S be a ring homomorphism where F is a field. Then f is injective. Proof. ker f is a proper ideal in F , so ker f = 0. Theorem 2.2.5. M is a maximal ideal () R=M is a field. 58 Proof. The 4th iso. thm. says 9 a bijection between the ideals of R containing M and the ideals of R=M. ) 9I s.t. M⊆I⊆R () 9J s.t. f0}⊆J⊆R=M. ie. M is not maximal () R=M is not a field. Definition 2.2.6. An ideal P⊆R is called a prime ideal if ab 2 P implies a 2 P or b 2 P. Theorem 2.2.7. P is a prime ideal () R=P is an integral domain. Proof. ): Suppose P is a prime ideal. If [xy] = [x][y] = 0 in R=P then xy 2 P, so either x 2 P or y 2 P. ie. either [x] = 0 or [y] = 0. Thus R=P has no zero divisors. (: Suppose R=P is an integral domain. If xy 2 P then [x][y] = 0 in R=P, so [x] = 0 or [y] = 0. ie. either x 2 P or y 2 P. Corollary 2.2.8. A maximal ideal is a prime ideal. Proof. A field is an integral domain. Notation: a j b means 9c s.t. b = ac (say a divides b). Proposition 2.2.9. In an integral domain, if a j b and b j a then b = ua for some unit u. Proof. a j b ) b = ua for some u 2 R. b j a ) a = vb for some v 2 R. ) b = ua = uvb, and since b is not a zero divisor, 1 = uv. Thus, u is a unit. Definition 2.2.10. q is called a greatest common divisor of a and b if: 1. q j a and q j b, and 2. If c also satisfies c j a; c j b then c j q. Notation: q = (a; b) means q is the greatest common divisor of a and b. We say a and b are relatively prime if (a; b) = 1. Proposition 2.2.11. Let R be an integral domain. If q = (a; b) and q0 = (a; b) then q0 = uq for some unit u. Conversely, if q = (a; b) and q0 = uq where u is a unit then q0 = (a; b). Proof. Let q = (a; b). If q0 = (a; b) then q0 j q and q j q0 so q0 = uq for some unit u. Conversely, if q0 = uq for some unit u then q0 j q so q0 j a and q0 j b. Also q j q0 so whenever c j a and c j b, c j q so c j q0. 59 Definition 2.2.12. A non-unit p =6 0 2 R is called a prime if p j ab ) p j a or p j b.