Prime \(M\)-ideals, \(M\)-prime submodules, \(M\)-prime radical and \(M\)-Baer's lower nilradical of modules. (Q2868507)

All rings in this paper are associative with identity and all modules are unitary. Throughout the paper \(_RM\) is a fixed left \(R\)-module and the category \(\sigma[M]\) is defined to be the full subcategory of \(R\)-Mod that contains all modules \(_RX\) such that \(X\) is isomorphic to a submodule of an \(M\)-generated module. Let \(R\) be a ring. In this paper for a left \(R\)-module \(X\), the authors introduce the notions of \(M\)-prime submodule, \(M\)-semiprime submodule of \(X\), prime \(M\)-ideal of \(M\) and semiprime \(M\)-ideal of \(M\) in such a way that in the case \(M=R\), the notion of an \(R\)-prime submodule (resp. \(R\)-semiprime submodule) reduces to the familiar definition of a prime submodule (resp. semiprime submodule) and the notion of an \(R\)-ideal (resp. prime \(R\)-ideal) of \(_RR\) reduces to the familiar definition of an ideal (resp. prime ideal) of \(R\). The authors also introduce other concepts such as \(M\)-\(m\)-system sets, \(M\)-\(n\)-system sets, \(M\)-prime radical and \(M\)-Baer's lower nilradical of modules which are natural yet too technical to be included in this review. The authors establish relationships between these notions and describe their basic properties. For example, they show that prime \(M\)-ideals play a role similar to that of prime two-sided ideals in the ring \(R\).NEWLINENEWLINE The module \(_RX\) in \(\sigma[M]\) is said to be finitely \(M\)-generated if there exists an epimorphism \(f\colon M^n\to X\), for some positive integer \(n\). It is said to be finitely \(M\)-annihilated if there exists a monomorphism \(g\colon M/\text{Ann}_M(X)\to X^m\) for some positive integer \(m\). Also, the module \(_RM\) is said to satisfy condition H if every finitely \(M\)-generated module is finitely \(M\)-annihilated. The authors prove that if \(M\) satisfies condition H and \(\Hom_R(M,X)\neq 0\) for all modules \(X\) in \(\sigma[M]\), then there is a one-to one correspondence between isomorphism classes of indecomposable \(M\)-injective modules in \(\sigma[M]\) and prime \(M\)-ideals of \(M\).NEWLINENEWLINE In the final section of the paper, the authors present some interesting results concerning the prime \(M\)-ideals, the \(M\)-prime submodules and the \(M\)-prime radical of Artinian modules. For example, they show that if \(X\) is a Noetherian \(R\)-module and every \(M\)-prime submodule of \(X\) is virtually maximal, then \(X/\text{rad}_M(X)\) is an Artinian \(R\)-module, where a proper submodule \(P\) of an \(R\)-module \(M\) is virtually maximal if the factor module \(M/P\) is a direct sum of isomorphic simple modules and \(\text{rad}_M(X)\) denotes the \(M\)-prime radical of \(X\).