Rings of functions which are discontinuous on a finite set with countable range

TL;DR

This paper studies a specific ring of real-valued functions with finite discontinuities and countable range, exploring its algebraic structure, filters, ideals, and a Gelfand-Kolmogoroff type theorem.

Contribution

It introduces and analyzes the structure of $(\mathcal{Z}_c)_F$-filters and ideals, and establishes a Gelfand-Kolmogoroff theorem analogue for this function ring.

Findings

Characterization of $(\mathcal{Z}_c)_F$-filters and ideals

Conditions for $C_c(X)_F$ to be a Baer-ring and regular ring

Analysis of the zero divisor graph on $C_c(X)_F$

Abstract

Consider the ring $C_c(X)_F$ of real valued functions which are discontinuous on a finite set with countable range. We discuss $(\mathcal{Z}_c)_F$-filters on $X$ and $(\mathcal{Z}_c)_F$-ideals of $C_c(X)_F$. We establish an analogous version of Gelfand-Kolmogoroff theorem in our setting. We prove some equivalent conditions when $C_c(X)_F$ is a Baer-ring and a regular ring. Lastly, we talk about the zero divisor graph on $C_c(X)_F$.