Multiplication modules and homogeneous idealization

Majid M. Ali

Abstract: All rings are commutative with identity and all modules are unital. Let $R$ be a ring, $M$ an $R$-module and $R\left( M\right)$, the idealization of $M$. Homogeneous ideals of $R\left(M\right)$ have the form $I${\tiny (+)}$N$ where $I$ is an ideal of $R$, $N$ a submodule of $M$ and $IM\subseteq N$. The purpose of this paper is to investigate how properties of a homogeneous ideal $I${\tiny (+)}$N$ of $R\left(M\right)$ are related to those of $I$ and $N$. We show that if $M$ is a multiplication $R$-module and $I${\tiny (+)}$N$ is a meet principal (join principal) homogeneous ideal of $R\left(M\right)$ then these properties can be transferred to $I$ and $N$. We give some conditions under which the converse is true. We also show that $I${\tiny (+)}$N$ is large (small) if and only if $N$ is large in $M$ ($I$ is a small ideal of $R$).

Keywords: multiplication module, meet principal module, join principal submodule, large submodule, small submodule, idealization

Classification (MSC2000): 13C13, 13C05, 13A15

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