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ON REGULAR MODULES

Part of: Homological methods

Published online by Cambridge University Press:  14 June 2011

HUANYIN CHEN  and
W. K. NICHOLSON
HUANYIN CHEN
Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, PR China (email: [email protected])
W. K. NICHOLSON*
Affiliation:
Department of Mathematics, University of Calgary, Calgary, Canada T2N 1N4 (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Several characterizations are given of (Zelmanowitz) regular modules among the torsionless modules, the locally projective modules, the nonsingular modules, and modules where certain submodules are pure. Along the way, a version of the unimodular row lemma for torsionless modules is given, and it is shown that a regular ring is left self-injective if and only if every nonsingular left module is regular.

Keywords

regular moduleslocally projective modulestorsionless modulesnonsingular modulesself-injective ringspure submodules

MSC classification

Secondary: 16E50: von Neumann regular rings and generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 280 - 287
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The research of the first author was supported by the Natural Science Foundation of Zhejiang Province (Y6090404), and the research of the second author was supported by NSERC (Canada) Grant A8075.

References

[1]Azumaya, G., ‘Some characterizations of regular modules’, Publ. Mat. 34 (1990), 241–248.CrossRefGoogle Scholar
[2]Goodearl, K. R., Von Neumann Regular Rings, 2nd edn (Krieger, Malabar, FL, 1991).Google Scholar
[3]Nicholson, W. K. and Yousif, M. F., Quasi-Frobenius Rings, Tracts in Mathematics, 158 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[4]Zelmanowitz, J. M., ‘Endomorphism rings of torsionless modules’, J. Algebra 5 (1967), 325–341.CrossRefGoogle Scholar
[5]Zelmanowitz, J. M., ‘Regular modules’, Trans. Amer. Math. Soc. 163 (1972), 341–355.CrossRefGoogle Scholar
[6]Zimmermann-Huisgen, B., ‘Pure submodules of direct products of free modules’, Math. Ann. 224 (1976), 233–245.CrossRefGoogle Scholar