On semiprime modules with chain conditions. (Q2923092)

Let \(R\) be an associative ring with identity, \(M\) a right \(R\)-module and \(S=\mathrm{End}_R(M)\). For any submodule \(X\) of \(M\), \(I_X\) denotes the set of all \(f\in S\) such that the image of \(f\) is contained in \(X\). This set is an ideal of \(S\) if \(X\) is a fully invariant submodule of \(M\). It is shown that, for \(M\) a right \(R\)-module, if \(X\) is a minimal prime submodule of \(M\), then \(I_X\) is a minimal prime ideal of \(S\) and if \(P\) is a minimal prime ideal of \(S\), then \(I_X=P\) for some minimal submodule of \(M\). For \(M\) a right \(R\)-module which is a self-generator and \(X\) a fully invariant submodule of \(M\), several equivalent conditions are found for \(X\) to be semiprime (i.e. an intersection of prime submodules).NEWLINENEWLINE A submodule (resp. a fully invariant submodule) \(X\) of a right \(R\)-module \(M\) is called an \(M\)-annihilator (resp. a full \(M\)-annihilator) if \(X=\mathrm{Ker}(I)=\bigcap_{f \in I}\mathrm{Ker}(f)\) for some subset (resp. ideal) \(I\) of \(S\). If a full \(M\)-annihilator \(X\neq M\) and there are no full \(M\)-annihilators between \(X\) and \(M\), then \(X\) is said to be maximal. For a quasi-projective semiprime finitely generated right \(R\)-module which is a self-generator and a proper fully invariant right submodule \(X\) of \(M\), \(X\) is shown to be a maximal full \(M\)-annihilator if and only if \(S\) is a minimal prime submodule and a full \(M\)-annihilator if and only if \(X\) is a prime submodule and a full \(M\)-annihilator.NEWLINENEWLINE The bi-module \(_SM_R\) is said to have bi-uniform dimension \(n\) (denoted \(\dim_SM_R\)) if there is a bi-essential submodule \(V\) of \(_SM_R\) such that \(V\) is a direct sum of bi-uniform submodules. For a quasi-projective, finitely generated right \(R\)-module which is a self-generator with finite bi-uniform dimension \(n\), it is proved that \(\dim{_SS_S}=n\). Conversely, if \(\dim{_SS_S}=n<\infty\), then \(\dim{_SM_S}=n\). On the other hand, \(\dim{_SM_S}=\infty\) if and only if \(_SM_S\) contains an infinite direct sum of nonzero bi-submodules.NEWLINENEWLINE A bi-submodule \(X\) of the bi-module \(_SM_S\) is said to be a bi-complement of \(_SM_R\) if there exists a bi-submodule \(Y\) of \(_SM_R\) such that \(X\) is a bi-complement of \(Y\) in \(_SM_R\). A bi-module \(_SM_R\) is shown to have infinite bi-uniform dimension if and only if there exists an infinite strictly ascending (or descending) chain of bi-complements in \(_SM_R\). For \(M\) a quasi-projective, semiprime, finitely generated right \(R\)-module which is a self-generator, \(M\) is shown to have finite bi-uniform dimension if and only if \(M_R\) has ACC on full \(M\)-annihilators if and only if \(M_R\) has DCC on full \(M\)-annihilators if and only if \(_SM_R\) has ACC on bi-complements if and only if \(_SM_R\) has DCC on bi-complements.