Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules

Abstract

Abstract. Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M -prime (resp. M -semiprime) submodule of X such that in the case M = R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M -m-system sets, M -n-system sets, M -prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M , called "prime M -ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and Hom R (M, X) = 0 for all modules X in the category σ[M ], then there is a one-to-one correspondence between isomorphism classes of indecomposable M -injective modules in σ[M ] and prime M -ideals of M . Also, we investigate the prime M -ideals, M -prime submodules and M -prime radical of Artinian modules.