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.

一个框架是一个完整的格子，其中相遇分布在任意连接上。设\(\tau \)是帧L的子帧，使得\(\tau \) 的每个元素在L 中都有一个补码，然后\((L, \tau )\)，简而言之\(L_{ \tau }\)，据说是一个拓扑框架。令\({\mathcal {R}}L_\tau \)是拓扑框架\(L_{ \tau }\)上的实连续函数环。我们定义P -topoframes 并证明\(L_{\tau }\)是P -topoframe 当且仅当\({\mathcal {R}}L_{\tau }\)是一个规则环当且仅当它是一个\(\aleph _0\)-自注射环。我们定义了极不连接的拓扑框，并表明\(L_{\tau }\)是一个极不连接的拓扑框，当且仅当\(\tau \)是一个极不连接的框。对于完全规则的拓扑框架\(L_\tau \)，表明\(L_\tau \)是极不连接的拓扑框架当且仅当\({\mathcal {R}}L_\tau \)是 Baer ring 当且仅当它是一个CS环。最后，我们证明了一个完全规则的拓扑框架\(L_\tau \)是一个极不连通的P拓扑框架，当且仅当\({\mathcal {R}}L_\tau \)是一个自注入环。