Prime Ideals in Function Rings

Abstract

In this chapter, the author aims to call attention to some algebraic properties of function rings and to some interesting results that can be derived by studying these algebraic properties in depth. He focuses on the structure of the prime ideals in C(X). The first prime ideals in C(X) that were studied were, of course, the maximal ones. The key feature of the ring A(G) that is needed is that each compact zero-set of G is also the zero-set of some function in A(G). As a consequence, there is a one-to-one correspondence between the prime z-filters on G that converge to the identity element, 0, and the prime z-ideals in A(G) that contain the ideal O. The author calls attention to an entirely different situation where prime ideals play a significant role, namely, the question of automatic continuity of homomorphisms of commutative Banach algebras.