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

## TL;DR

This paper investigates the structure of ideals in the ring of Baire one functions, exploring duality with zero sets, characterizing maximal ideals, and analyzing residue class rings, revealing algebraic properties and relationships in topological spaces.

## Contribution

It provides a detailed duality framework between ideals and zero sets in $B_1(X)$, characterizes fixed and free maximal ideals, and studies residue class rings, advancing understanding of $B_1(X)$'s algebraic structure.

## Key findings

- $B_1(X)$ is a Gelfand ring but not Noetherian.
- Fixed maximal ideals are fully described for certain spaces.
- Residue class rings are analyzed with respect to real and hyper real maximal ideals.

## 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 $B_1^*(X)$ are obtained. Though free maximal ideals of $B_1(X)$ and those of $B_1^*(X)$ 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 $B_1(X)$.