Rings of Continuous Functions

Abstract

This work consists only of a survey of [1]. In this talk, we study rings of continuous functions and their ideals and z-lters. Throughout this study, C(X) denotes all continuous, real-valued functions on a topological space X and C (X) denotes all bounded functions in C(X). C(X) is a commutative ring with identity and also C (X) is a subring of C(X). Here, we use f 1 for multiplicative inverse of f and f for inverse of a mapping f. Also, f_g means supremum of f and g, likely f^ g means of f and g. Now, we give a theorem about invariants of homomorphisms: Let C(Y ) and C(X) be isomorphic, then C (Y ) and C (X) are isomorphic. We denote Z(f) as f (0) and denote the set fZ(f)j f2 Ig as Z[I] for an ideal I of C(X). An ideal I of C(X) is called z-ideal if Z(f)2 Z[I] implies f2 I. We give some theorems about characterization of z-ideal in C(X): Theorem. ([1], Theorem 2.8) Every z-ideal in C(X) is a intersection of prime ideals. Theorem. ([1], Theorem 2.9) For any z-ideal in C(X), the followings are equivalent: