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].