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Large subdirect products of projective modules. (English) Zbl 0664.16019
If \(\{M_{\alpha}\}_{\alpha \in A}\) is a family of left R-modules, let \(\sup p(x)=\{\alpha \in A|\) \(x_{\alpha}\neq 0\}\) be the support of x where \(x=(x_{\alpha})\in \prod_{\alpha \in A}M_{\alpha}\). If \(| X|\) denotes the cardinality of a set X and \(\aleph\) is an infinite cardinal number, the \(\aleph\)-product of the family \(\{M_{\alpha}\}_{\alpha \in A}\) is defined to be \(\prod_{\alpha \in A}^{\aleph}M_{\alpha}=\{x\in \prod_{\alpha \in A}M_{\alpha}|\) \(| \sup p(x)| <\aleph \}\). Since the direct product and the direct sum of the family \(\{M_{\alpha}\}_{\alpha \in A}\) are special cases of the \(\aleph\)-product, it is natural to study the flatness (projectivity) of the \(\aleph\)-products when each member of the family \(\{M_{\aleph}\}_{\alpha \in A}\) is flat (projective). The main results of the paper characterize the flatness (projectivity) of the \(\aleph\)-product in terms of the ideal structure of the ring. The author uses the concepts of \(\aleph\)-coherent and \(\aleph\)-perfect rings to generalize well-known results on coherent and perfect rings.
Reviewer: P.E.Bland

16D40 Free, projective, and flat modules and ideals in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings
Full Text: DOI
[1] DOI: 10.1090/S0002-9947-1960-0157984-8
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[3] DOI: 10.1090/S0002-9947-1960-0120260-3
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