Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 47, No. 1, pp. 249270 (2006) 

Multiplication modules and homogeneous idealizationMajid M. AliDepartment of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, P.C. 123 AlKhod, Sultanate of Oman, email: mali@squ.edu.omAbstract: 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 Full text of the article:
Electronic version published on: 9 May 2006. This page was last modified: 4 Nov 2009.
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