Geodesic
QF-3 endomorphism rings of Σ-quasi-projective modules†
Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 215-220

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Pardo, José L. Gómez; González, Nieves Rodríguez. QF-3 endomorphism rings of Σ-quasi-projective modules†. Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 215-220. doi: 10.1017/S0017089500007254
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[1] 1.Albu, T. and Năstăsescu, C., Relative finiteness in module theory (Marcel Dekker, 1984). Google Scholar

[2] 2.Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, 1974). Google Scholar | DOI

[3] 3.Colby, R. R. and Rutter, E. A. Jr, QF-3 rings with zero singular ideal, Pacific J. Math. 28 (1969), 303–308. Google Scholar | DOI

[4] 4.Faith, C. and Walker, E. A., Direct sum representations of injective modules, J. Algebra 5 (1967), 203–221. Google Scholar | DOI

[5] 5.Hernández, J. L. García and Pardo, J. L. Gómez, Self-injective and PF endomorphism rings, Israeli. Math., 58 (1987), 324–350. Google Scholar | DOI

[6] 6.Kasch, F., Modules and rings (Academic Press, 1982). Google Scholar

[7] 7.Page, S. S., FPF endomorphism rings with applications to QF-3 rings, Comm. Algebra 14 (1986), 423–435. Google Scholar | DOI

[8] 8.Ringel, C. M. and Tachikawa, H., QF-3 rings, J. Reine Angew. Math. 272 (1975), 49–72. Google Scholar

[9] 9.Rutter, E. A. Jr, PF and QF-3 rings, Arch. Math. 19 (1968), 608–611. Google Scholar | DOI

[10] 10.Tachikawa, H., On left QF-3 rings, Pacific J. Math. 32 (1970), 255–268. Google Scholar | DOI

[11] 11.Wisbauer, R., Decomposition properties in module categories Acta Univ. Carolinae Math. Phys. 26 (1985), 57–68. Google Scholar

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