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\({\mathcal F}\)-products of injective, flat, and projective modules. (English) Zbl 0727.16002
Let R be a ring with identity element, let \({\mathcal F}\) be a filter on a set A, and let \(\{M_{\alpha}|\alpha\in A\}\) be a set of unital right R-modules. Define an equivalence relation \(\sim\) on the direct product \(\prod_{\alpha \in A}M_{\alpha}\) as follows: \((m_{\alpha})\sim (m_{\alpha}')\) if and only if \(\{\alpha\in A|\) \(m_{\alpha}=m'_{\alpha}\}\in {\mathcal F}\). The equivalence class of 0 is called the \({\mathcal F}\)-product of the \(M_{\alpha}'s\) and is denoted by \(\prod^{{\mathcal F}}M_{\alpha}\). For an infinite cardinal number \(\sigma\), \({\mathcal F}_{\sigma}=\{B\subseteq A|| A-B| <\sigma \}\) is a filter. R is right Noetherian if and only if, for every non- principal filter \({\mathcal F}\) on any index set A and every family \(\{E_{\alpha}|\alpha\in A\}\) of injective right R-modules, \(\prod^{{\mathcal F}}E_{\alpha}\) is injective. Every strictly ascending chain of right ideals of R has strictly less than \(\sigma\) terms (\(\sigma\) regular, \(\sigma >\aleph_ 0)\) if and only if the \({\mathcal F}_{\sigma}\)-product of any family of injective right R-modules is injective. Necessary and sufficient conditions are given for any \({\mathcal F}_{\sigma}\)-product of flat right R-modules to be flat. R is left coherent if and only if, for every filter \({\mathcal F}\) on a set A and every family \(\{M_{\alpha}|\alpha\in A\}\) of flat right R-modules, \(\prod^{{\mathcal F}}M_{\alpha}\) is flat. Necessary and sufficient conditions are given for any \({\mathcal F}_{\sigma}\)-product (\(\sigma\) regular) of projective right R-modules to be projective. Finally, R is right perfect and left coherent if and only if, for every filter \({\mathcal F}\) on a set A and every family \(\{P_{\alpha}|\alpha\in A\}\) of projective right R-modules, \(\prod^{{\mathcal F}}P_{\alpha}\) is projective.

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16P40 Noetherian rings and modules (associative rings and algebras)
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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