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. We further investigate when is $C(X)_\mathcal{P}$ closed under uniform limit, for various choices of $\mathcal{P}$. The concept of $\mathcal{F}P$-space was introduced in \cite{GGT2018} and in that paper, the authors established a result stating that every $P$-space is a $\mathcal{F}P$-space. We show that this theorem might fail if $X$ is not Tychonoff and we provide a suitable counterexample to prove our assertion.