Direct sum of local modules with extending factor modules (Q1371123)

A module \(M\) (over any associative ring with 1) is called an extending module (or CS-module) provided every submodule of \(M\) is essential in a direct summand of \(M\). If \(M\) is an extending module and for all direct summands \(A\) and \(B\) of \(M\) with \(A\cap B=0\), the submodule \(A\oplus B\) is also a direct summand of \(M\), then \(M\) is called quasi-continuous. Rings over which cyclic (right) modules are quasi-continuous have been studied, e.g., by \textit{V. K. Goel} and \textit{S. K. Jain} [Commun. Algebra 6, No. 1, 59-73 (1978; Zbl 0368.16010)]\ and \textit{B. L. Osofsky} and \textit{P. F. Smith} [J. Algebra 139, No. 2, 342-354 (1991; Zbl 0737.16001)]. In the present paper, the authors study modules \(M\) satisfying the property that every factor module of \(M\) is extending, under the additional hypothesis that \(M\) is a direct sum of local modules (a module is called local if it contains a largest proper submodule). As the main result of the paper, the authors prove that if \(M=\bigoplus_{i\in I}M_i\) is a direct sum of local modules, then the following conditions are equivalent: (a) Every factor module of \(M\) is extending; (b) \(\bigoplus_{i\in I}M_i/X_i\) is extending for all submodules \(X_i\) of \(M_i\); (c) \(M\) is extending and every factor module of \(M/\text{Soc}(M)\) is quasi-continuous. Moreover, under the above equivalent conditions, the decomposition \(M=\bigoplus_{i\in I}M_i\) complements direct summands in the sense of \textit{F. W. Anderson} and \textit{K. R. Fuller} [Rings and Categories of Modules (Springer-Verlag 1974; 301.16001)]. (The reader is referred also to the reviewer's recent paper [J. Algebra 197, No. 2, 449-467 (1997; Zbl 0887.16005)]\ where necessary and sufficient conditions are given for an arbitrary indecomposable decomposition of a module to complement direct summands).