Remarks on the ring B 1 (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 fiite set and B 1( X) be the ring of all realvalued Baire one functions on X. We show that any member of B 1( X) is a zero divisor
or a unit. We give an algebraic characterization of X when every subset of X is a G δ-set.
We give an algebraic characterization of X when for every p ∈ X, there exists f ∈ B 1( X)
such that {p} = f^{− 1}(0) and we give some topological characterizations of minimal ideals,
essential ideals and socle of B 1(X). Some relations between C(X) F, B 1(X) and some
interesting function rings on X are studied and investigated. We show that B 1(X) is a
regular ring if and only if every countable intersection of sets functionally open can be
represented as a countable union of sets functionally closed.