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

Consider the ring Formula: see text comprising real-valued functions exhibiting discontinuity on finite sets with a countable range. Within this context, we explore the Formula: see text-filters on Formula: see text and Formula: see text-ideals in Formula: see text. We establish an adaptation of the Gelfand–Kolmogoroff theorem within this framework and present various equivalent conditions when Formula: see text conforms to the properties of a Baer-ring or a regular ring. Additionally, we delve into the zero divisor graph analysis within Formula: see text.