Rings of functions whose closure of discontinuity set is in an ideal of closed sets

Abstract: Let $\mathcal{P}$ be an ideal of closed subsets of a topological space $X$. Consider the ring, $C(X)\mathcal{P}$ of real valued functions on $X$ whose closure of discontinuity set is a member of $\mathcal{P}$. We investigate the ring properties of $C(X)\mathcal{P}$ for different choices of $\mathcal{P}$, such as the $\aleph_0$-self injectivity and regularity of the ring, if and when the ring is Artinian and/or Noetherian. The concept of $\mathcal{F}P$-space was introduced by Z. Gharabaghi, M. Ghirati and A. Taherifar in 2018 in a paper published in Houston Journal of Mathematics. In this paper, they established a result stating that every $P$-space is a $\mathcal{F}P$-space. We furnish that this theorem might fail if $X$ is not Tychonoff and we provide a suitable counter example to prove our assertion.