Notes on Ring Homomorphisms and Ideals

16. Ring Homomorphisms and Ideals Definition 16.1. Let φ : R −→ S be a function between two rings. We say that φ is a ring homomorphism if for every a and b ∈ R, φ(a + b) = φ(a) + φ(b) φ(a · b) = φ(a) · φ(b), and in addition φ(1) = 1. Note that this gives us a category, the category of rings. The objects are rings and the morphisms are ring homomorphisms. Just as in the case of groups, one can define automorphisms. Example 16.2. Let φ : C −→ C be the map that sends a complex number to its complex conjugate. Then φ is an automorphism of C. In fact φ is its own inverse. Let φ : R[x] −→ R[x] be the map that sends f (x) to f (x + 1). Then φ is an automorphism. Indeed the inverse map sends f (x) to f (x − 1). By analogy with groups, we have Definition 16.3. Let φ : R −→ S be a ring homomorphism. The kernel of φ, denoted Ker φ, is the inverse image of zero. As in the case of groups, a very natural question arises. What can we say about the kernel of a ring homomorphism? Since a ring homo­ morphism is automatically a group homomorphism, it follows that the kernel is a normal subgroup. However since a ring is an abelian group under addition, in fact all subgroups are automatically normal. Definition-Lemma 16.4. Let R be a ring and let I be a subset of R. We say that I is an ideal of R and write I < R if I is a an additive subgroup of R and for every a ∈ I and r ∈ R, we have ra ∈ I and ar ∈ I. Let φ : R −→ S be a ring homorphism and let I be the kernel of φ. Then I is an ideal of R. Proof. We have already seen that I is an additive subgroup of R. Suppose that a ∈ I and r ∈ R. Then φ(ra) = φ(r)φ(a) = φ(r)0 = 0. Thus ra is in the kernel of φ. Similarly for ar. 1 D As before, given an additive subgroup H of R, we let R/H denote the group of left cosets of H in R. Proposition 16.5. Let R be a ring and let I be an ideal of R, such that I = R. Then R/I is a ring. Furthermore there is a natural ring homomor­ phism u : R −→ R/I which sends r to r + I. Proof. As I is an ideal, and addition in R is commutative, it follows that R/I is a group, with the natural definition of addition inherited from R. Further we have seen that φ is a group homomorphism. It remains to define a multiplication in R/I. Given two left cosets r+I and s+I in R/I, we define a multiplication in the obvious way, (r + I)(s + I) = rs + I. In fact this is forced by requiring that u is a ring homorphism. As before the problem is to check that this is well-defined. Suppose that r' + I = r + I and s' + I = s + I. Then we may find i and j in I such that r' = r + i and s' = s + j. We have r' s' = (r + i)(s + j) = rs + is + rj + ij. As I is an ideal, is + rj + ij ∈ I. It follows that r' s' + I = rs + I and multiplication is well-defined. The rest is easy to check. D As before the quotient of a ring by an ideal is a categorical quotient. Theorem 16.6. Let R be a ring and I an ideal not equal to all of R. Let u : R −→ R/I be the obvious map. Then u is universal amongst all ring homomorphisms whose kernel contains I. That is, suppose φ : R −→ S is any ring homomorphism, whose kernel contains I. Then there is a unique ring homomomorphism ψ : R/I −→ S, which makes the following diagram commute, φ R u ψ /S / R/I. Theorem 16.7. (Isomorphism Theorem) Let φ : R −→ S be a homo­ morphism of rings. Suppose that φ is onto and let I be the kernel of φ. 2 Then S is isomorphic to R/I. Example 16.8. Let R = Z. Fix a non-zero integer n and let I consist of all multiples of n. It is easy to see that I is an ideal of Z. The quotient, Z/I is Zn the ring of integers modulo n. Definition-Lemma 16.9. Let R be a commutative ring and let a ∈ R be an element of R. The set I = (a) = { ra | r ∈ R }, is an ideal and any ideal of this form is called principal. Proof. We first show that I is an additive subgroup. Suppose that x and y are in I. Then x = ra and y = sa, where r and s are two elements of R. In this case x + y = ra + sa = (r + s)a. Thus I is closed under addition. Further −x = −ra = (−r)a, so that I is closed under inverses. It follows that I is an additive subgroup. Now suppose that x ∈ I and that s ∈ R. Then sx = s(ra) = (sr)a ∈ I. It follows that I is an ideal. D Definition-Lemma 16.10. Let R be a ring. We say that u ∈ R is a unit, if u has a multiplicative inverse. Let I be an ideal of a ring R. If I contains a unit, then I = R. Proof. Suppose that u ∈ I is a unit of R. Then vu = 1, for some v ∈ R. It follows that 1 = vu ∈ I. Pick a ∈ R. Then a = a · 1 ∈ I. D Proposition 16.11. Let R be a division ring. Then the only ideals of R are the zero ideal and the whole of R. In particular if φ : R −→ S is any ring homomorphism then φ is injective. Proof. Let I be an ideal, not equal to {0}. Pick u ∈ I, u = 0. As R is a division ring, it follows that u is a unit. But then I = R. Now let φ : R −→ S be a ring homomorphism and let I be the kernel. Then I cannot be the whole of R, so that I = {0}. But then φ is injective. D 3 Example 16.12. Let X be a set and let R be a ring. Let F denote the set of functions from X to R. We have already seen that F forms a ring, under pointwise addition and multiplication. Let Y be a subset of X and let I be the set of those functions from X to R whose restriction to Y is zero. Then I is an ideal of F . Indeed I is clearly non-empty as the zero function is an element of I. Given two functions f and g in F , whose restriction to Y is zero, then clearly the restriction of f + g to Y is zero. Finally, suppose that f ∈ I, so that f is zero on Y and suppose that g is any function from X to R. Then gf is zero on Y . Thus I is an ideal. Now consider F/I. I claim that this is isomorphic to the space of functions G from Y to R. Indeed there is a natural map from F to G which sends a function to its restriction to Y , f −→ f |Y . It is clear that the kernel is I. Thus the result follows by the Isomor­ phism Theorem. As a special case, one can take X = [0, 1] and R = R. Let Y = {1/2}. Then the space of maps from Y to R is just a copy of R. Example 16.13. Let R be the ring of Gaussian integers, that is, those complex numbers of the form a + bi. Let I be the subset of R consisting of those numbers such 2|a and 2|b. I claim that I is an ideal of R. In fact suppose that a + bi ∈ I and c + di ∈ I. Then (a + bi) + (c + di) = (a + c) + (b + d)i. As a and c are even, then so is a + c and similarly as b and d are even, then so is b + d. Thus I is closed under addition. Similarly I is closed under inverses. Now suppose that a + bi ∈ I and r = c + di is a Gaussian integer. Then (c + di)(a + bi) = (ac − bd) + (ad + bc)i. As a and b are even, so are ac − bd and ad + bc and so I is an ideal. 4