Regular substructures of Hom. (Q2426117)

For two right \(R\)-modules \(A_R\) and \(M_R\) the following structures are considered: \(H=\Hom_R(A,M)\), \(S=\text{End}(M_R)\), \(T=\text{End}(A_R)\), where \(_SH_T\) is an \(S\)-\(T\)-bimodule. Some important subsets of \(_SH_T\) are studied (as \(S\)-\(T\)-submodules) and the relations of them with the substructures of \(A\) and \(M\) are investigated. An element \(f\in H\) is called regular if there exists \(g\in\Hom_R(M,A)\) such that \(fgf=f\). For such element we have: \(S_f\) is \(S\)-projective and \(S_f\subseteq^\otimes{_SH}\); \(_fT\) is \(T\)-projective and \(_fT\subseteq^\otimes H_T\). The following \(S\)-\(T\)-submodule of \(H\) is defined and studied: \(\text{Reg}(A,M)=\{f\in H\mid SfT\) is regular\} (the largest regular \(S\)-\(T\)-submodule of \(_SH_T\)). If \(\text{Reg}(A,M)\neq 0\) and \(A\) is indecomposable, then \(T\) is a division ring. The relation between the regular elements of \(H\) and regular elements of \(\Hom_R(M,A)\) is mentioned (the transfer rule: \((f,g)\mapsto(gfg,f)\)). Some interesting results are formulated for relative regularity (\(U\)-regularity, semi-regularity).