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On countable \(\Sigma\)-CS modules. (English) Zbl 1125.16002

Huynh, Dinh V. (ed.) et al., Algebra and its applications. Proceedings of the international conference, Athens, OH, USA, March 22–26, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3842-3/pbk). Contemporary Mathematics 419, 1-6 (2006).
Let \(R\) be an associative ring with identity. All modules are unital right \(R\)-modules. A module \(M\) is called CS if every submodule is essential in a direct summand of \(M\) and it is called (countably) \(\Sigma\)-CS if every direct sum of copies of \(M\) is (countably) CS. The authors give the following definition: given a module \(M\) and an ordinal \(\gamma\), \(M\) is said to have \(\gamma\)-ACC on monomorphisms if every directed system of proper monomorphisms \(\{f_{\alpha\beta}\colon M_\alpha\to M_\beta\}_{\alpha\leq\beta<\delta}\) with \(M_\alpha\cong M\) for every \(\alpha<\delta\) satisfies that \(\delta<\gamma\).
Let \(M=\bigoplus_{i\in I}M_i\) be a direct sum of uniform modules, suppose that \(M\) is countably \(\Sigma\)-CS, and denote by \(\omega_1\) the first uncountable ordinal. Among other results, the authors establish the equivalence of the following statements: (1) \(M\) is \(\Sigma\)-CS; (2) Each \(M_i\) has \(\omega_1\)-ACC on monomorphisms; (3) Each \(M_i\) has \(\omega_1\)-ACC on submodules isomorphic to \(M_i\).
For the entire collection see [Zbl 1104.16300].

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D50 Injective modules, self-injective associative rings
16L30 Noncommutative local and semilocal rings, perfect rings
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