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.
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Acknowledgements
Thanks are due to the referee for helpful comments that have improved the readability of this paper and for providing Corollary 4.10.
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Estaji, A.A., Abedi, M. On regularity and injectivity of the ring of real-continuous functions on a topoframe. Algebra Univers. 82, 60 (2021). https://doi.org/10.1007/s00012-021-00753-2
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Keywords
- Topoframe
- Frame
- Ring of all real-continuous functions on a topoframe
- P-topoframe
- Extremally disconnected topoframe
- Injective
- \(\aleph _0\)-self-injective
- Regular ring
Mathematics Subject Classification
- 06D22
- 06F25
- 54C30
- 16E50
- 16D50
- 54G05