环$C(X)_\mathcal{P}$的代数性质
Algebraic properties of the ring $C(X)_\mathcal{P}$
AI总结：
该文研究实值函数环$C(X)_\mathcal{P}$的代数性质，通过各类理想与拓扑刻画$\mathcal{P}P$-空间及几乎$\mathcal{P}P$-空间，证明环中五类环概念一致，还给出素理想与本质理想深度为零的$\mathcal{P}$条件。
摘要：
我们的目标是研究$X$上的实值函数环$C(X)_\mathcal{P}$的若干代数性质，该环中函数的不连续点集的闭包属于闭集理想。我们利用$C(X)_\mathcal{P}$的$z$-理想与本质理想刻画了$\mathcal{P}P$-空间，还利用$C(X)_\mathcal{P}$的$z^0$-理想以及一种比$X$上原有拓扑更细的拓扑刻画了几乎$\mathcal{P}P$-空间。我们推导出$C(X)_F$（文献\cite{GGT2018}）（对应地，$T'(X)$（文献\cite{A2010}））的每个极大理想都是$z^0$-理想。我们证明了在环$C(X)_\mathcal{P}$中，clean环、弱clean环、半clean环、几乎clean环与exchange环的概念是一致的。在论文末尾，我们还利用若干深度为零的理想刻画了$\mathcal{P}P$-空间与几乎$\mathcal{P}P$-空间，并给出了$\mathcal{P}$满足的一个条件，在此条件下$C(X)_\mathcal{P}$的素理想与本质理想的深度为零。
英文摘要：
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. $T'(X)$ \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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
