Homomorphisms, ideals, and factor rings

Abstract

Abstract It is often the case that two rings for which we already have some information are in fact related to each other. For instance the ring 71./nl of integers mod n is in some sense derived from the ring 71. of integers. We shall now study such connections between rings. This is done by means of homomorphisms, which are the appropriate type of function from one ring to another. One reason for doing this is that it helps us to organize our examples of rings by establishing natural connections between some of them. Closely related to homomorphisms are ideals which are certain subsets of a ring. The properties of the two are intimately connected. This theory then allows us to construct new examples of rings with certain specified properties. We shall give an important application of this type in Chapters 14 and 15, where we shall show how to construct a finite field with pn elements for any choice of the prime number p and positive integer n.