Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 47, No. 1, pp. 249-270 (2006)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



Multiplication modules and homogeneous idealization

Majid M. Ali

Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, P.C. 123 Al-Khod, Sultanate of Oman, e-mail:

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

Full text of the article:

Electronic version published on: 9 May 2006. This page was last modified: 4 Nov 2009.

© 2006 Heldermann Verlag
© 2006–2009 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition