Extending modules with injective or semisimple summands
A module \(M\) is called an extending module if every submodule of \(M\) is essential in a direct summand. A natural question that arises in the study of extending modules is to determine necessary and sufficient conditions for a direct sum of extending modules to be extending. The paper under review addresses this question, and gives satisfactory answers in some special cases. For example, necessary and sufficient conditions are obtained for a direct sum of an extending module and an injective module to be extending, or for a direct sum of a module with the finite exchange property and a semisimple module to be extending. These conditions involve some relative injectivity property between the summands. Finally, it is proved that a ring \(R\) satisfies the property that every direct sum of an extending right \(R\)-module and a semisimple right \(R\)-module is extending if and only if \(R/\text{Soc}(R_R)\) is a right Noetherian right \(V\)-ring.
scientific article; zbMATH DE number 1308163
scientific article