Strong regular and weak regular rings and modules (Q1897937)

A module \(M\) is strongly regular if every submodule of \(M\) which is not contained in the Jacobson radical \(J(M)\) is a direct summand of \(M\). The author first shows that if \(M\neq J(M)\) is strongly regular then \(J(M)\) is superfluous in \(M\). A module \(M\) is weakly regular if every submodule of \(M\) which is not contained in \(J(M)\) contains a non-radical cyclic direct summand of \(M\). The rest of the paper characterizes weakly regular modules.