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RING HOMOMORPHISMS


Definitions: Ring Homomorphism, Ring Isomorphism

A ring homomorphism f from a ring R to a ring S is a mapping from R to S that preserves the two ring operations; that is, for all a, b in R
f(a + b) = f(a) + f(b) and f(ab) = f(a) f(b)

A ring homomorphism that is both one-to-one and onto is called a ring isomorphism. An isomorphism is used to show that two rings are identical; a homomorphism is used to simplify a ring while retaining some of its features. Here are some examples:

(1) For any positive integer n, the mapping k ® k mod n is a ring homomorphism from Z onto Zn. This mapping is called the natural homomorphism from Z to Zn.
(2...