Journal of the
Korean Mathematical Society JKMS

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 ``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 $M$ satisfies condition $H$ (defined later) and ${\rm Hom}_R(M,X)\neq 0$ 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

MSC numbers: 16S38, 16D50, 16D60, 16N60