Closed weak supplemented modules. (Q863759)

A module \(M\) is called closed weak supplemented if for any closed submodule \(N\) of \(M\), there is a submodule \(K\) of \(M\) such that \(M=N+K\) and \(K\cap N\) is small in \(N\). Several properties of closed weak supplemented modules are studied. For example, any direct summand of a closed weak supplemented module is also closed weak supplemented. A ring \(R\) is closed weak supplemented if the module \(R_R\) is closed weak supplemented. It is shown that a ring \(R\) is closed weak supplemented if and only if the \(n\)-by-\(n\) matrix ring \(M_n(R)\) over \(R\) is closed weak supplemented for any positive integer \(n\).