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

TL;DR

This paper surveys the ring-theoretic properties of the ring of real-continuous functions on topoframes, extending concepts like P-spaces and extremally disconnected spaces to this setting, and characterizes regularity and injectivity in terms of topoframe properties.

Contribution

It extends classical topological concepts to topoframes and characterizes algebraic properties of the function ring in terms of topoframe structures.

Findings

The ring is $eth$-Kasch for P-topoframes.

Regularity of the ring is equivalent to $eth$-selfinjectivity.

The ring is a Bear ring if and only if the topoframe is extremally disconnected.

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.