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 B 1 (X) be the ring of all real-valued 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 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 cozero sets of continuous functions can be represented as a countable union of zero sets of continuous functions.