Algebraic properties of the ring $C(X{)}_{\mathcal{P}}$

Abstract

Our aim is to study certain algebraic properties of the ring $C(X{)}_{\mathcal{P}}$ of real-valued functions on $X$ whose closure of discontinuity set is in an ideal of closed sets. We characterize $\mathcal{P}P$-spaces using $z$-ideals and essential ideals of $C(X{)}_{\mathcal{P}}$ and also almost $\mathcal{P}P$-spaces using $z^0$-ideals of $C(X{)}_{\mathcal{P}}$ and a topology finer than the original topology on $X$. We deduce that each maximal ideal of $C(X{)}_F$ \cite{GGT2018} (resp. $T^{\prime }(X)$ \cite{A2010}) is a $z^0$-ideal. We establish that the notions of clean ring, weakly clean ring, semiclean ring, almost clean ring and exchange ring coincide in the ring $C(X{)}_{\mathcal{P}}$. End of this paper, we also characterize $\mathcal{P}P$-spaces and almost $\mathcal{P}P$-spaces using certain ideals having depth zero. We exhibit a condition on $\mathcal{P}$ under which prime and essential ideals of $C(X{)}_{\mathcal{P}}$ have depth zero.