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RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

Published online by Cambridge University Press:  30 March 2012

PINAR AYDOĞDU ,
NOYAN ER  and
NİL ORHAN ERTAŞ
PINAR AYDOĞDU
Affiliation:
Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey e-mail: paydogdu@hacettepe.edu.tr
NOYAN ER
Affiliation:
Department of Mathematics, University of Rio Grande, Rio Grande, OH 45674, USA e-mail: noyaner@yahoo.com
NİL ORHAN ERTAŞ
Affiliation:
Department of Mathematics, Karabük University, 78050 Karabük, Turkey e-mail: orhannil@yahoo.com
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Abstract

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Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

Keywords

Primary 16D1016D7016P2016P40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 605 - 617
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

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