On regularity and injectivity of the ring of real-continuous functions on a topoframe

Abstract

A frame is a complete lattice in which the meet distributes over arbitrary joins. Let $\tau$ be a subframe of a frame L such that every element of $\tau$ has a complement in L, then $(L, \tau )$, briefly $L_{ \tau }$, is said to be a topoframe. Let ${\mathcal {R}}L_\tau$ be the ring of real-continuous functions on a topoframe $L_{ \tau }$. We define P-topoframes and show that $L_{\tau }$ is a P-topoframe if and only if ${\mathcal {R}}L_{\tau }$ is a regular ring if and only if it is a $\aleph _0$-self-injective ring. We define extremally disconnected topoframes and show that $L_{\tau }$ is an extremally disconnected topoframe if and only if $\tau$ is an extremally disconnected frame. For a completely regular topoframe $L_\tau$, it is shown that $L_\tau$ is an extremally disconnected topoframe if and only if ${\mathcal {R}}L_\tau$ is a Baer ring if and only if it is a CS-ring. Finally, we prove that a completely regular topoframe $L_\tau$ is an extremally disconnected P-topoframe if and only if ${\mathcal {R}}L_\tau$ is a self-injective ring.