Every \(\aleph_1\)-\(\Sigma\)-CS module is \(\Sigma\)-CS.
Let \(R\) be an associative ring. A right \(R\)-module \(M\) is called CS when every submodule of \(M\) is essential in a direct summand. The modules \(M\) such that every direct sum of copies of \(M\) is CS are called \(\Sigma\)-CS modules and, more generally, \(M\) is \(\aleph\)-\(\Sigma\)-CS, where \(\aleph\) is an infinite cardinal, whenever every direct sum of \(\aleph\) copies of \(M\) is CS. \(\Sigma\)-CS modules are direct sums of uniform modules and, more generally, it was also known that every \(\aleph_1\)-\(\Sigma\)-CS module is a direct sum of uniforms, but the question whether every \(\aleph_1\)-\(\Sigma\)-CS module is already \(\Sigma\)-CS remained open. In this paper, the author gives a positive solution to this problem. In order to obtain this result, the author proves the following interesting fact: If \(M\) is a uniform module such that each local direct summand of a countable direct sum of copies of \(M\) is a direct summand, then \(M\) has local endomorphism ring.
scientific article; zbMATH DE number 2104460