The Bass-Papp theorem and some related results
The well-known Bass-Papp Theorem states that a ring \(R\) is right Noetherian if and only if every direct sum of injective right \(R\)-modules is injective. A right \(R\)-module \(M\) is called a \(CS\)-module if every submodule of \(M\) is essential in a direct summand of \(M\). If \(M\) is a \(CS\)-module and for all direct summands \(K\) and \(L\) of \(M\) with \(K\cap L=0\), the submodule \(K\oplus L\) is also a direct summand of \(M\), then \(M\) is called quasi-continuous. \(CS\)-modules and quasi-continuous modules generalize injective modules, and the paper under review deals with the question of characterizing the rings over which direct sums of \(CS\)-modules (or quasi-continuous modules) are again \(CS\) (respectively, quasi-continuous). It is shown that the direct sum of any two quasi-continuous right \(R\)-modules is quasi-continuous if and only if \(R\) is semisimple Artinian. The similar question about when the class of \(CS\)-modules is closed under direct sums seems to be more difficult, and has no definitive answer in the paper. Among other partial results, it is shown that if the direct sum of any two \(CS\) right \(R\)-modules is \(CS\), then the ring \(R\) must be semi-artinian. Furthermore, if \(R\) has finitely generated right socle, then the direct sum of any two \(CS\) right \(R\)-modules is \(CS\) if and only if every right \(R\)-module is \(CS\), if and only if \(R\) is (left and right) Artinian serial and \(J(R)^2=0\).
scientific article; zbMATH DE number 1054656