Prime M-ideals, M-prime submodules, M-prime radical and M-Baer's lower nilradical of modules

Let M be a fixed left R-module. For a left R-module X, weintroduce the notion of M-prime (resp. M-semiprime) submodule of Xsuch that in the case M = R, it coincides with prime (resp. semiprime)submodule of X. Other concepts encountered in the general theory areM-m-system sets, M-n-system sets, M-prime radical and M-Baer’s lowernilradical of modules. Relationships between these concepts and basicproperties are established. In particular, we identify certain submodulesof M, called “primeM-ideals”, that play a role analogous to that of prime(two-sided) ideals in the ring R. Using this definition, we show that if Msatisfies condition H (defined later) and HomR(M,X) ≠ 0 for all mod-ules X in the category σ[M], then there is a one-to-one correspondencebetween isomorphism classes of indecomposable M-injective modules inσ[M] and prime M-ideals of M. Also, we investigate the prime M-ideals,M-prime submodules and M-prime radical of Artinian modules.