Aleph-nought-continuous regular rings
Although aleph-nought-continuous (von Neumann) regular rings do not possess the Type I, II, III structural decomposition theory available for continuous or self-injective regular rings, they nonetheless have a good structure theory (based on factor rings rather than summands). For right and left aleph-nought-continuous regular rings, this was initiated by \textit{D. Handelman} [Adv. Math., Suppl. Stud., Vol. 4, 171-196 (1979; Zbl 0511.46054)] and continued by him together with \textit{D. Higgs} and \textit{J. Lawrence} [J. Lond. Math. Soc., II. Ser. 21, 193-202 (1980; Zbl 0449.06013)] and \textit{J. Lawrence} and the reviewer [Mem. Am. Math. Soc. 234 (1980; Zbl 0435.16005)]. The reviewer later extended most of this structure theory to directly finite right aleph-nought-continuous regular rings [Pac. J. Math. 100, 105-122 (1982; Zbl 0453.16007)]. In the present paper, the author pushes on to structure theory for an arbitrary right aleph-nought-continuous regular ring \(R\). His key result (comparable to the directly finite/purely infinite decomposition for a continuous regular ring) is that the set \(I\) of those \(x\in R\) for which \(R\oplus xR\) embeds in \(R\) is a two-sided ideal such that \(R/I\) is directly finite and right aleph-nought-continuous. Using various lifting techniques, he is then able to remove the directly finite hypothesis from a number of earlier results. For example, if \(A\) and \(B\) are finitely generated projective right \(R\)-modules such that \(nA\cong nB\) (or \(nA\) embeds in \(nB\)) for some positive integer \(n\), then \(A\cong B\) (or \(A\) embeds in \(B\)). Two open problems from the reviewer's book [Von Neumann regular rings (1979; Zbl 0411.16007)] are answered (affirmatively) as well.
directly finite right aleph-nought-continuous regular rings
scientific article; zbMATH DE number 4002304