PRIME M-IDEALS, M-PRIME SUBMODULES, M-PRIME RADICAL AND M-BAER'S LOWER NILRADICAL OF MODULES

Title & Authors
PRIME M-IDEALS, M-PRIME SUBMODULES, M-PRIME RADICAL AND M-BAER'S LOWER NILRADICAL OF MODULES
Beachy, John A.; Behboodi, Mahmood; Yazdi, Faezeh;

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-$\small{m}$-system sets, M-$\small{n}$-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 "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 M satisfies condition H (defined later) and $\small{Hom_R(M,X){\neq}0}$ for all modules X in the category $\small{{\sigma}[M]}$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in $\small{{\sigma}[M]}$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.
Keywords
prime submodules;prime M-ideal;M-prime submodule;M-prime radical;M-injective module;
Language
English
Cited by
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