Endoprime modules. (Q812802)

Let \(M\) be a right \(R\)-module over a ring \(R\) with identity. Then \(R\) is a left \(S\)-module over the endomorphism ring \(S=\text{End}(M_R)\). A nonzero right \(R\)-module \(M\) is called endoprime if every nonzero fully invariant submodule of \(M_R\) is faithful as a left \(S\)-module. The class of endoprime modules properly contains the class of prime modules in the sense of \textit{J. Zelmanowitz} [J. Algebra 25, 554-574 (1973; Zbl 0258.16002)]. The purpose of this paper is to study endoprime modules, and some general properties of endoprime \(R\)-modules are obtained. For example, the authors prove that \(M_R\) is endoprime if and only if the left \(S\)-module \(_SM\) is prime; if \(M_R\) is endoprime then \(S\) is a prime ring; if \(R\) is commutative and \(M_R\) is endoprime then \(M_R\) prime. Moreover, the structure of the ring \(S\) is determined when \(M\) is endoprime and, in addition, \(M\) is suitably restricted.