### Remarks on the ring B 1 (X )

#### Abstract

Let

on

or a unit. We give an algebraic characterization of

We give an algebraic characterization of

such that

essential ideals and socle of

interesting function rings on

regular ring if and only if every countable intersection of sets functionally open can be

represented as a countable union of sets functionally closed.

*X*be a nonempty topological space,*C*(*X*)*F*be the set of all real-valued functionson

*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 divisoror 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 someinteresting function rings on

*X*are studied and investigated. We show that*B*1(*X*) is aregular ring if and only if every countable intersection of sets functionally open can be

represented as a countable union of sets functionally closed.

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