CS-modules and annihilator conditions. (Q1415175)

Recall that a module \(M\) is said to be extending or a CS-module if every (essentially) closed submodule of \(M\) is a direct summand of \(M\). Let \(R\) and \(S\) be rings and \(_SM_R\) be a bimodule. This paper studies the connection between the CS-condition on the module \(M_R\) and the annihilator condition that \(S=\ell_S(A)+\ell_S(B)\) for any closed submodule \(A_R\) of \(M_R\) and a complement \(B_R\) of \(A_R\), where \(\ell_S(X)\) denotes the left annihilator in \(S\) of a subset \(X\) of \(M\). For example, after more general results it is shown that, taking \(S=\text{End}(M_R)\), then \(M_R\) is extending if and only if, for every closed submodule \(A\) of \(M_R\) there exists a complement \(B\) of \(A\) in \(M\) satisfying \(S=\ell_S(A)+\ell_S(B)\). The authors also study what they call \(W^*\)-bimodules, namely those \(_SM_R\) with the condition that \(r_M\ell_S(A)=A\) for every closed submodule \(A\) of \(M_R\), where \(r_M(Y)\) denotes the right annihilator in \(M\) of any \(Y\subseteq R\). This leads to characterizations of continuous and quasi-continuous modules.