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ON A RECENT GENERALIZATION OF SEMIPERFECT RINGS

Part of: Local rings and generalizations

Published online by Cambridge University Press:  01 October 2008

ENGIN BÜYÜKAŞIK  and
CHRISTIAN LOMP
ENGIN BÜYÜKAŞIK
Affiliation:
Department of Mathematics, Izmir Institute of Technology, 35430, Urla, Izmir, Turkey (email: enginbuyukasik@iyte.edu.tr)
CHRISTIAN LOMP*
Affiliation:
Departamento de Matemâtica, Pura da Faculdade de Ciências da Universidade do Porto, R. Campo Alegre 687, 4169-007 Porto, Portugal (email: clomp@fc.up.pt)
*
✉For correspondence; e-mail: clomp@fc.up.pt
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Abstract

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In a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding’s claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of ‘local’ for submodules.

Keywords

semiperfect ringssupplemented modulesgeneralized supplemented modulesw-local modules

MSC classification

Secondary: 16L30: Noncommutative local and semilocal rings, perfect rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 317 - 325
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The first author thanks the Scientific and Technical Research Council of Turkey (TÜBITAK) for financial support. The second author was supported by Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto (CMUP).

References

[1]Alizade, R., Bilhan, G. and Smith, P. F., ‘Modules whose maximal submodules have supplements’, Comm. Algebra 29 (2001), 2389–2405.CrossRefGoogle Scholar
[2]Al-Takhman, K., Lomp, C. and Wisbauer, R., ‘τ-complemented and τ-supplemented modules’, Algebra Discrete Math. 3 (2006), 1–15.Google Scholar
[3]Bass, H., ‘Finitistic dimension and a homological generalization of semiprimary rings’, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
[4]Bessenrodt, C., Brungs, H. H. and Törner, G., Right Chain Rings. Part 1, Schriftenreihe des Fachbereiches Mathematik, 181 (Universität Duisburg, Duisburg, 1990).Google Scholar
[5]Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R., ‘Lifting modules’, in: Supplements and Projectivity in Module Theory, Frontiers in Mathematics, 406 (Birkhäuser, Basel, 2006).Google Scholar
[6]Dubrovin, N. I., ‘Chain domains’, Mosc. Univ. Math. Bull. 35(2) (1980), 56–60.Google Scholar
[7]Facchini, A., ‘Module theory’, in: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics, 167 (Birkhäuser, Basel, 1998).Google Scholar
[8]Gerasimov, V. N. and Sakhaev, I. I., ‘A counterexample to two conjectures on projective and flat modules’, Sibirsk. Mat. Zh. 25 (1984), 31–35 (Russian).Google Scholar
[9]Kasch, F. and Mares, E. A., ‘Eine Kennzeichnung semi-perfekter Moduln’, Nagoya Math. J. 27 (1966), 525–529.CrossRefGoogle Scholar
[10]Puninski, G., ‘Projective modules over the endomorphism ring of a biuniform module’, J. Pure Appl. Algebra 188 (2004), 227–246.CrossRefGoogle Scholar
[11]Wang, Y. and Ding, N., ‘Generalized supplemented modules’, Taiwanese J. Math 10 (2006), 1589–1601.CrossRefGoogle Scholar
[12]Ware, R., ‘Endomorphism rings of projective modules’, Trans. Amer. Math. Soc. 155 (1971), 233–256.CrossRefGoogle Scholar
[13]Wisbauer, R., Foundations of Modules and Rings (Gordon and Breach, London, 1991).Google Scholar