Open Access
september 2018 On injectivity of the ring of real-valued continuous functions on a frame
Ali Akbar Estaji, Mostafa Abedi
Bull. Belg. Math. Soc. Simon Stevin 25(3): 467-480 (september 2018). DOI: 10.36045/bbms/1536631239

Abstract

We give characterizations of $P$-frames and extremally disconnected $P$-frames based on ring-theoretic features of the ring of continuous real- valued functions on a frame $L$, i.e. $\mathcal RL$. It is shown that $L$ is a $P$-frame if and only if $\mathcal RL$ is an $\aleph_0$-self-injective ring. Consequently for pseudocompact frames if $\mathcal RL$ is $\aleph_0$-self-injective, then $L$ is finite. We also prove that $L$ is an extremally disconnected $P$-frame iff $\mathcal{R}L$ is a self-injective ring iff $\mathcal{R}L$ is a Baer regular ring iff $\mathcal{R}L$ is a continuous regular ring iff $\mathcal{R}L$ is a complete regular ring.

Citation

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Ali Akbar Estaji. Mostafa Abedi. "On injectivity of the ring of real-valued continuous functions on a frame." Bull. Belg. Math. Soc. Simon Stevin 25 (3) 467 - 480, september 2018. https://doi.org/10.36045/bbms/1536631239

Information

Published: september 2018
First available in Project Euclid: 11 September 2018

zbMATH: 06861554
MathSciNet: MR3852680
Digital Object Identifier: 10.36045/bbms/1536631239

Subjects:
Primary: 06D22 , 13A30 , 16D25 , 16D50 , ‎54C30 , 54G05

Keywords: $\aleph_0$-self-injective , $P$-frame , Extremally disconnected frame , Ring of continuous real-valued functions on a frame , self-injective

Rights: Copyright © 2018 The Belgian Mathematical Society

Vol.25 • No. 3 • september 2018
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