Abstract Algebra Theory and Applications - Computer Science ...

14.3 RING HOMOMORPHISMS AND IDEALS 2391 < b < n. Since 0 = n1 = (ab)1 = (a1)(b1) and there are no zero divisorsin D, either a1 = 0 or b1 = 0. Hence, the characteristic of D must be lessthan n, which is a contradiction. Therefore, n must be prime.□14.3 Ring Homomorphisms and IdealsIn the study of groups, a homomorphism is a map that preserves the operationof the group. Similarly, a homomorphism between rings preservesthe operations of addition and multiplication in the ring. More specifically,if R and S are rings, then a ring homomorphism is a map φ : R → Ssatisfyingφ(a + b) = φ(a) + φ(b)φ(ab) = φ(a)φ(b)for all a, b ∈ R. If φ : R → S is a one-to-one and onto homomorphism, thenφ is called an isomorphism of rings.The set of elements that a ring homomorphism maps to 0 plays a fundamentalrole in the theory of rings. For any ring homomorphism φ : R → S,we define the kernel of a ring homomorphism to be the setker φ = {r ∈ R : φ(r) = 0}.Example 13. For any integer n we can define a ring homomorphismφ : Z → Z n by a ↦→ a (mod n). This is indeed a ring homomorphism,sinceandφ(a + b) = (a + b) (mod n)= a (mod n) + b (mod n)= φ(a) + φ(b)φ(ab) = ab (mod n)= a (mod n) · b (mod n)= φ(a)φ(b).The kernel of the homomorphism φ is nZ.Example 14. Let C[a, b] be the ring of continuous real-valued functionson an interval [a, b] as in Example 4. For a fixed α ∈ [a, b], we can define