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Consider the ring $C_c(X)_F$ of real valued functions which are discontinuouson 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 analogousversion of Gelfand-Kolmogoroff theorem in our setting. We prove some equivalentconditions when $C_c(X)_F$ is a Baer-ring and a regular ring. Lastly, we talkabout the zero divisor graph on $C_c(X)_F$.

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