Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(1): 83-102

Online first article November 8, 2018      Printed January 31, 2019

https://doi.org/10.4134/BKMS.b180122

Copyright © The Korean Mathematical Society.

A generalization of multiplication modules

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

Supported by: This work was supported by the grant UNAM-DGAPA-PAPIIT IN100517.

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