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QF-3 rings and torsion theories

Published online by Cambridge University Press:  09 April 2009

José L. Gómez Pardo  and
Nieves Rodríguez González
José L. Gómez Pardo
Affiliation:
Departamento de Matemáticas, Universidad de Murcia30001 Murcia, Spain
Nieves Rodríguez González
Affiliation:
Departamento de Matemáticas, Universidad de Murcia30001 Murcia, Spain
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Abstract

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In this paper, the rings which have a torsion theory τ with associated torsion radical τ such that R/t(R) has a minimal τ-torsionfree cogenerator are studied. When τ is the trivial torsion theory these are precisely the left QF-3 rings. For τ = τL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on τL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 251 - 261
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules, Graduate Texts in Math., 13 (Springer-Verlag, New York, 1974).CrossRefGoogle Scholar
[2]Baccella, G., ‘The structure of QF-3 rings with zero singular ideal’, Comm. Algebra 15 (1987), 1393–1446.CrossRefGoogle Scholar
[3]Colby, R. R. and Rutter, E. A. Jr, ‘QF-3 rings with zero singular ideal’, Pacific J. Math. 28 (1969), 303–308.CrossRefGoogle Scholar
[4]Hernández, J. L. García and Gómez Pardo, J. L., ‘V-rings relative to Gabriel topologies’, Comm. Algebra 13 (1985), 59–83.CrossRefGoogle Scholar
[5]Hernández, J. L. García and Pardo, J. L. Gómez, ‘Self-injective and PF endomorphism rings’, Israel J. Math. 58 (1987), 324–350.CrossRefGoogle Scholar
[6]García Hernández, J. L., and Pardo, J. L. Gómez, ‘closed submodules of free modules over the endomorphism ring of a quasi-injective module’, Comm. Algebra 16 (1988), 115–137.CrossRefGoogle Scholar
[7]Golan, J. S., Torsion theories, Longman, Harlow, 1986.Google Scholar
[8]Pardo, J. L. Gómez, ‘Embedding cyclic and torsion-free modules in free modules’, Arch. Math. 44 (1985), 503–510.CrossRefGoogle Scholar
[9]Masaike, K., ‘Semiprimary QF-3 quotient rings’, Comm. Algebra 11 (1983), 377–389.CrossRefGoogle Scholar
[10]Miller, R. W. and Teply, M. L., ‘The descending chain condition relative to a torsion theory’, Pacific J. Math. 83 (1979), 207–218.CrossRefGoogle Scholar
[11]Ringel, C. M. and Tachikawa, H., ‘QF-3 rings’, J. Reine Angew. Math. 272 (1975), 49–72.Google Scholar
[12]Rutter, E. A. Jr, ‘A characterization of QF-3 rings’, Pacific J. Math. 51 (1974), 533–536.CrossRefGoogle Scholar
[13]Rutter, E. A. Jr, ‘QF-3 rings with ascending chain condition on annihilators’, J. Reine Angew. Math. 277 (1975), 40–44.Google Scholar
[14]Rutter, E. A. Jr, ‘Dominant modules and finite localizations’, Tôhoku Math. J. 27 (1975), 225–239.CrossRefGoogle Scholar
[15]Stenström, B., Rings of quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[16]Tachikawa, H., ‘On left QF-3 rings’, Pacific J. Math. 32 (1970), 255–268.CrossRefGoogle Scholar