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

A. Deb Ray, Atanu Mondal

Abstract

This paper explores the duality between ideals of the ring B1(X) of all real valued Baire one functions on a topological space X and typical families of zero sets, called ZB-filters, on X. As a natural outcome of this study, it is observed that B1(X) is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of B1(X), complete descriptions of the fixed maximal ideals of both B1(X) and B1* (X) are obtained. Though free maximal ideals of B1(X) and those of B1* (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 T1 space X, free maximal ideals of B1(X) are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of B1(X) modulo an ideal, with special emphasize on real and hyper real maximal ideals of B1(X).


Keywords

ZB- filter; ZB -ultrafilter; ZB -ideal; fixed ideal; free ideal; residue class ring; real maximal ideal; hyper real maximal ideal

Subject classification

26A21; 54C30; 13A15; 54C50.

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References

A. Deb Ray and A. Mondal, On rings of Baire one functions, Applied Gen. Topol. 20, no. 1 (2019), 237-249. https://doi.org/10.4995/agt.2019.10776

J. P. Fenecios and E. A. Cabral, On some properties of Baire-1 functions, Int. Journal of Math. Analysis 7, no. 8 (2013), 393-402. https://doi.org/10.12988/ijma.2013.13035

L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2

J. R. Munkres, Topology, Second edition, Pearson Education, Delhi, 2003.

L. Vesely, Characterization of Baire-one functions between topological spaces, Acta Universitatis Carolinae. Mathematica et Physica 33, no. 2 (1992), 143-156.

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Universitat Politècnica de València

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