Alahmadi, Adel N.; Al-Hazmi, Husain S.; Guil Asensio, Pedro A. 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]. Reviewer: Iuliu Crivei (Cluj-Napoca) 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 Keywords:uniform modules; \(\Sigma\)-CS modules; direct summands; direct sums; monomorphisms PDFBibTeX XMLCite \textit{A. N. Alahmadi} et al., Contemp. Math. 419, 1--6 (2006; Zbl 1125.16002)