What are the suﬃcient and necessary conditions on a topological space X such that the ring C(X) satisﬁes certain ring theoretical assumptions?
Among diﬀerent types of rings that arise as the examples and counterexamples in a ring and module theory one of the most important places is reserved by the rings of continuous functions with values in some prescribed ring. One of the most popular types of such rings are continuous real-valued functions C(X) deﬁned on a topological space X. Here and thereafter we assume that real numbers R are equipped with standard topology and all rings are commutative with nontrivial identity.
Deﬁning operations of addition and multiplication between two given functions pointwise we equip set of continuous real-valued functions C(X) and set of continuous bounded real-valued functions C∗(X) with a ring structure. Naturally one can ask a question: “what are the suﬃcient and necessary conditions on a topological space X such that the ring C (X ) satisﬁes certain ring theoretical assumptions?”
The main part of the following is a survey about the possible answers to such questions. We start with the basic notions from topology that we will use extensively throughout the article.