Some ring-theoretic properties of the ring of $\mathcal{R}{L}_{\tau }$

Abstract

The aim of this article is to survey ring-theoretic properties of Kasch, the regularity and the injectivity of the ring of real-continuous functions on a topoframe $L_{\tau }$, i.e., $\mathcal{R}{L}_{\tau }$. In order to study these properties, the concept of $P$-spaces and extremally disconnected spaces are extend to topoframes. For a $P$- topoframe $L_{\tau }$, the ring $\mathcal{R}{L}_{\tau }$ is ${\aleph }_0$-Kasch ring. $P$- topoframes are characterized in terms of ring-theoretic properties of the regularity and injectivity of the ring of real-continuous functions on a topoframe. It follows from these characterizations that the ring $\mathcal{R}{L}_{\tau }$ is regular if and only if it is ${\aleph }_0$-selfinjective. For a completely regular topoframe $L_{\tau }$, we show that $\mathcal{R}{L}_{\tau }$ is a Bear ring if and only if it is a $CS$-ring if and only if $L_{\tau }$ is extremally disconnected and also prove that it is selfinjective ring if and only if $L_{\tau }$ is an extremally disconnected $P$-topoframe.