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

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.