Endo-principally quasi-Baer modules. (Q2788559)

Let \(R\) be a ring with \(1\), \(M\) a right \(R\)-module and \(S=\text{End}_R(M)\). Then \(M\) is called an endo-principally quasi-Baer module (= endo-p.q.-Baer) if for every \(m\in M\), the left annihilator of \(Sm\) in \(S\) is \(Se\) for some \(e^2=e\in S\). A ring \(R\) is called left p.q.-Baer if the left annihilator of any left principal ideal of \(R\) is \(Re\) for some \(e^2=e\in R\).NEWLINENEWLINE The following statements are equivalent: (1) \(R\) is a left p.q.-Baer ring. (2) Every free right \(R\)-module is an endo-p.q.-Baer module. (3) Every projective right \(R\)-module is an endo-p.q.-Baer module.NEWLINENEWLINE A module \(M\) is called an endo-principally extending module if for every \(m\in M\), there is an idempotent \(e\in S\) such that the submodule of \(M\) spanned by \(Sm\) is essential in \(eM\), an (\(FT\)-\(K\))-nonsingular module if for any invariant ideal \(I\) of \(S\) such that the right annihilator of \(I\) in \(M\), \(r_M(I)\) essential in \(eM\), \(r_M(I)=eM\), and (\(FT\)-\(K\))-cononsingular if for any invariant submodule \(N\) of a direct summand of \(M\) and \(N'\) an invariant submodule of \(N\) such that \(\varphi(N')\neq 0\) for every \(\varphi\in\text{End}_R(N)\) implies \(N'\) essential in \(N\). Then every (\(FT\)-\(K\))-nonsingular endo-principal extending module is endo-p.q.-Baer, and every (\(FT\)-\(K\))-cononsingular endo-p.q.-Baer module is an endo-principally extending module. Moreover, it is shown that \(\text{End}_R(M)\) is a left p.q.-Baer ring if \(M\) is an endo-p.q.-Baer module and \(\text{End}_R(M)\) has no infinite set of nonzero orthogonal right semicentral idempotents \(e\) (that is, \(er=ere\) for each \(r\in R\)).