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ON THE OSOFSKY–SMITH THEOREM*

Published online by Cambridge University Press:  24 June 2010

SEPTIMIU CRIVEI ,
CONSTANTIN NĂSTĂSESCU  and
BLAS TORRECILLAS
SEPTIMIU CRIVEI
Affiliation:
Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania e-mail: crivei@math.ubbcluj.ro
CONSTANTIN NĂSTĂSESCU
Affiliation:
Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania e-mail: cnastase@al.math.unibuc.ro
BLAS TORRECILLAS
Affiliation:
Departamento de Álgebra y Análisis, Universidad de Almería, 04071 Almería, Spain e-mail: btorreci@ual.es
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Abstract

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We recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

Keywords

16D5016S90
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue A: Rings and Modules in Honour of Patrick F. Smith's 65th Birthday , July 2010 , pp. 61 - 67
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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