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QF-3 endomorphism rings of Σ-quasi-projective modules†

Published online by Cambridge University Press:  18 May 2009

José L. Gómez Pardo  and
Nieves Rodríguez González
José L. Gómez Pardo
Affiliation:
Departamento de Algebra, Faculted de Ciencials Químicas y Matemáticals, Universidad de Murcia, 30001 Murcia, Spain
Nieves Rodríguez González
Affiliation:
Departamento de Algebra, Faculted de Ciencials Químicas y Matemáticals, Universidad de Murcia, 30001 Murcia, Spain
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A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 2 , May 1988 , pp. 215 - 220
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

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