Abstract Algebra Theory and Applications - Computer Science ...

14.4 MAXIMAL AND PRIME IDEALS 243Proof. Let K = ker φ. By the First Isomorphism Theorem for groups,there exists a well-defined group homomorphism η : R/K → ψ(R) definedby η(r+K) = ψ(r) for the additive abelian groups R and R/K. To show thatthis is a ring homomorphism, we need only show that η((r + K)(s + K)) =η(r + K)η(s + K); butη((r + K)(s + K)) = η(rs + K)= ψ(rs)= ψ(r)ψ(s)= η(r + K)η(s + K).Theorem 14.12 (Second Isomorphism Theorem) Let I be a subringof a ring R and J an ideal of R. Then I ∩ J is an ideal of I andI/I ∩ J ∼ = (I + J)/J.Theorem 14.13 (Third Isomorphism Theorem) Let R be a ring andI and J be ideals of R where J ⊂ I. ThenR/I ∼ = R/JI/J .Theorem 14.14 (Correspondence Theorem) Let I be a ideal of a ringR. Then S → S/I is a one-to-one correspondence between the set of subringsS containing I and the set of subrings of R/I. Furthermore, the ideals of Rcontaining I correspond to ideals of R/I.□14.4 Maximal and Prime IdealsIn this particular section we are especially interested in certain ideals ofcommutative rings. These ideals give us special types of factor rings. Morespecifically, we would like to characterize those ideals I of a commutativering R such that R/I is an integral domain or a field.A proper ideal M of a ring R is a maximal ideal of R if the idealM is not a proper subset of any ideal of R except R itself. That is, Mis a maximal ideal if for any ideal I properly containing M, I = R. Thefollowing theorem completely characterizes maximal ideals for commutativerings with identity in terms of their corresponding factor rings.