Ideals in B1(X) and residue class rings of B1(X) modulo an ideal

Abstract

<p>This paper explores the duality between ideals of the ring B<sub>1</sub>(X) of all real valued Baire one functions on a topological space X and typical families of zero sets, called Z<sub>B</sub>-filters, on X. As a natural outcome of this study, it is observed that B<sub>1</sub>(X) is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of B<sub>1</sub>(X), complete descriptions of the fixed maximal ideals of both B<sub>1</sub>(X) and B<sub>1</sub><sup>*</sup> (X) are obtained. Though free maximal ideals of B<sub>1</sub>(X) and those of B<sub>1</sub><sup>*</sup> (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<sub>1</sub> space X, free maximal ideals of B<sub>1</sub>(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<sub>1</sub>(X) modulo an ideal, with special emphasize on real and hyper real maximal ideals of B<sub>1</sub>(X).</p>