Remarks on the ring B1(X)

Abstract

Let X be a nonempty topological space, C(X)F be the set of all real-valued functions on X which are discontinuous at most on a finite set and B1(X) be the ring of all real-valued Baire one functions on X. We show that any member of B1(X) is a zero divisor or a unit. We give an algebraic characterization of X when for every p ? X, there exists f ? B1(X) such that {p} = f ?1(0) and we give some topological characterizations of minimal ideals, essential ideals and socle of B1(X). Some relations between C(X)F, B1(X) and some interesting function rings on X are studied and investigated. We show that B1(X) is a regular ring if and only if every countable intersection of cozero sets of continuous functions can be represented as a countable union of zero sets of continuous functions.