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 A generalization of multiplication modules Bull. Korean Math. Soc. 2019 Vol. 56, No. 1, 83-102 https://doi.org/10.4134/BKMS.b180122Published online 2019 Jan 31 Jaime Castro Perez, Jos\'{e} R\'{\i}os Montes, Gustavo Tapia S\'{a}nchez Calle del Puente 222, Tlalpan; Circuito Exterior, C. U.; Avenida del Charro 450 Norte, Partido Romero 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. Keywords : multiplication modules, prime modules, semiprime modules, Goldie modules MSC numbers : 16S90, 16D50, 16P50, 16P70 Full-Text :