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RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN*

Published online by Cambridge University Press:  24 June 2010

C. J. HOLSTON
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA e-mail: holston@math.ohiou.edu
S. K. JAIN
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, USA Department of Mathematics, King Abdulaziz University Jeddah, KSA e-mail: jain@math.ohiou.edu
A. LEROY
Affiliation:
Department of Mathematics, University of Artois, Rue J. Souvraz, 62300 Lens, France e-mail: andre.leroy@univ-artois.fr
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Abstract

R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = XT, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = ST, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

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