[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2013.50.6.1271

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

Beachy, John A. (Department of Mathematical Sciences Northern Illinois University)
Behboodi, Mahmood (Department of Mathematical Sciences Isfahan University of Technology, School of Mathematics Institute for Research in Fundamental Sciences (IPM))
Yazdi, Faezeh (Department of Mathematical Sciences Isfahan University of Technology)

Publication Information

Journal of the Korean Mathematical Society / v.50, no.6, 2013 , pp. 1271-1290 More about this Journal

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 "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 $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$ , then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ 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;

Citations & Related Records

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