On \(\aleph _ 0\)-injective regular rings
type: Item
instance of: scholarly article
title: On \(\aleph _ 0\)-injective regular rings
zbMATH Open document ID: 0594.16006
DOI: 10.1016/0022-4049(86)90073-3
author: Pere Ara
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: This note is concerned with (von Neumann) regular rings R which are right and left \(\aleph_ 0\)-injective (any homomorphism from a countably generated one-sided ideal of R to R can be extended to R). Although regular right and left self-injective rings are directly finite by an old result of Y. Utumi, this is not the case for regular right and left \(\aleph_ 0\)-injective rings, and the author finds general counterexamples among factor rings of directly infinite, prime, regular, right self-injective rings S. Namely, a proper factor ring S/I is right and left \(\aleph_ 0\)-injective (S/I is always directly infinite) if and only if I is a nonzero maximal ideal of S corresponding (with respect to the infinite dimension function introducd by the reviewer and A. K. Boyle) to the successor of a singular cardinal. A characterization of the right \(\aleph_ 0\)-continuous factor rings of S is also found. Another source of examples is the observation of \textit{D. E. Handelman} that a countably infinite direct product of copies of an arbitrary regular ring, modulo the corresponding direct sum, is always right and left \(\aleph_ 0\)-injective [Mich. Math. J. 28, 229-240 (1981; Zbl 0468.46045); pp. 237- 8].
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 3955137
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: regular right and left self-injective rings
zbMATH Keywords: regular right and left \(\aleph _ 0\)-injective rings
zbMATH Keywords: right \(\aleph _ 0\)-continuous factor rings
zbMATH Keywords: countably infinite direct product
Wikidata QID: Q114215399
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90073-3
OpenAlex ID: W2034694932
cites work: Artinian quotient rings of group rings
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cites work: Homomorphisms of C* algebras to finite AW* algebras
cites work: The structure of the infinite general linear group
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description: scientific article; zbMATH DE number 3955137