On non-\(M\)-singular modules with \((C_{11}^+)\) (Q1872918)

The author studies non-\(M\)-singular modules with the property that each submodule has a complement which is a direct summand in the module and with the summand intersection property, in the full subcategory of \(R\)-Mod whose objects are all \(R\)-modules subgenerated by \(M\). Conditions for an \(R\)-module to satisfy this condition are given. Special conditions are found in the case of the module \(N^n\), where \(N\) is a non-\(M\)-singular uniform right \(R\)-module; for example, this has the above properties if and only if \(\text{End}(N)\) is right semi-hereditary and \(N\) is a flat left End\((N)\)-module.