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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

##### MSC:
 16D40 Free, projective, and flat modules and ideals in associative algebras 16L30 Noncommutative local and semilocal rings, perfect rings
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##### References:
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