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Let M be a fixed left R-module. For a left R-module X, we introduce thenotion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R,which coincides with prime (resp. semiprime) submodule of X. Other conceptsencountered in the general theory are M-m-system sets, M-n-system sets, M-primeradical and M-Baer's lower nilradical of modules. Relationships between theseconcepts and basic properties are established. In particular, we identifycertain submodules of M, called "prime M-ideals", that play a role analogous tothat of prime (two-sided) ideals in the ring R. Using this definition, we showthat if M satisfes condition H (defined latter) and Hom_R(M,X)\neq 0$ for allmodules X in the category \sigma[M], then there is a one-to-one correspondencebetween isomorphism classes of indecomposable M-injective modules in \sigma[M]and prime M-ideals of M. Also, we investigate the prime M-ideals, M-primesubmodules and M-prime radical of Artinian modules.

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