Ideals in $B_1(X)$ and residue class rings of $B_1(X)$ modulo an ideal

Abstract

This paper explores the duality between ideals of the ring $B_1(X)$ of all real valued Baire one functions on a topological space $X$ and typical families of zero sets, called $Z_B$-filters, on $X$. As a natural outcome of this study, it is observed that $B_1(X)$ is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of $B_1(X)$, complete descriptions of the fixed maximal ideals of both $B_1(X)$ and $B_1^{\ast }(X)$ are obtained. Though free maximal ideals of $B_1(X)$ and those of $B_1^{\ast }(X)$ do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. It is proved here that for a perfectly normal $T_1$ space $X$, free maximal ideals of $B_1(X)$ are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of $B_1(X)$ modulo an ideal, with special emphasize on real and hyper real maximal ideals of $B_1(X)$.