Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules (original) (raw)
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, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are Mm 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 satisfes condition H (defined latter) 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 Minjective 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.
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Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1987