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DUO MODULES

Published online by Cambridge University Press:  06 December 2006

A. Ç. ÖZCAN ,
A. HARMANCI  and
P. F. SMITH
A. Ç. ÖZCAN
Affiliation:
Hacettepe University, Department of Mathematics 06532 Beytepe, Ankara TURKEY e-mail: [email protected], [email protected]
A. HARMANCI
Affiliation:
Hacettepe University, Department of Mathematics 06532 Beytepe, Ankara TURKEY e-mail: [email protected], [email protected]
P. F. SMITH
Affiliation:
University of Glasgow Department of Mathematics Glasgow G12 8QW Scotland, UK e-mail: [email protected]
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Abstract

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Let $R$ be a ring. An $R$-module $M$ is called a (weak) duo module provided every (direct summand) submodule of $M$ is fully invariant. It is proved that if $R$ is a commutative domain with field of fractions $K$ then a torsion-free uniform $R$-module is a duo module if and only if every element $k$ in $K$ such that $kM$ is contained in $M$ belongs to $R$. Moreover every non-zero finitely generated torsion-free duo $R$-module is uniform. In addition, if $R$ is a Dedekind domain then a torsion $R$-module is a duo module if and only if it is a weak duo module and this occurs precisely when the $P$-primary component of $M$ is uniform for every maximal ideal $P$ of $R$.

Keywords

16D9913C12 (13B20)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 48 , Issue 3 , September 2006 , pp. 533 - 545
Copyright
2006 Glasgow Mathematical Journal Trust