A generalization of multiplication modules
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For $M\in R$-Mod, $N\subseteq M$ and $L\in \sigma \left[ M \right] $ we consider the product $N_{M}L=\sum_{f\in {\rm Hom}_{R} ( M,L ) }f ( N ) $. A module $N\in \sigma \left[ M\right] $ is called an $M$-multiplication module if for every submodule $L$ of $N$, there exists a submodule $I$ of $M$ such that $L=I_{M}N$. We extend some important results given for multiplication modules to $M$-multiplication modules. As applications we obtain some new results when $M$ is a semiprime Goldie module. In particular we prove that $M$ is a semiprime Goldie module with an essential socle and $N$ $\in \sigma \left[ M\right] $ is an $M$ -multiplication module, then $N$ is cyclic, distributive and semisimple module. To prove these results we have had to develop new methods.
Abstract
For $M\in R$-Mod, $N\subseteq M$ and $L\in \sigma \left[ M \right] $ we consider the product $N_{M}L=\sum_{f\in {\rm Hom}_{R} ( M,L ) }f ( N ) $. A module $N\in \sigma \left[ M\right] $ is called an $M$-multiplication module if for every submodule $L$ of $N$, there exists a submodule $I$ of $M$ such that $L=I_{M}N$. We extend some important results given for multiplication modules to $M$-multiplication modules. As applications we obtain some new results when $M$ is a semiprime Goldie module. In particular we prove that $M$ is a semiprime Goldie module with an essential socle and $N$ $\in \sigma \left[ M\right] $ is an $M$ -multiplication module, then $N$ is cyclic, distributive and semisimple module. To prove these results we have had to develop new methods.
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