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UNIT-REGULAR MODULES

Part of: Homological methods

Published online by Cambridge University Press:  23 February 2017

H. CHEN ,
W. K. NICHOLSON  and
Y. ZHOU
H. CHEN
Affiliation:
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, PR China e-mail: [email protected]
W. K. NICHOLSON
Affiliation:
Department of Mathematics, University of Calgary, Calgary, T2N 1N4, Canada e-mail: [email protected]
Y. ZHOU
Affiliation:
Department of Mathematics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada e-mail: [email protected]
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Abstract

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In 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

MSC classification

Primary: 16E50: von Neumann regular rings and generalizations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 1 - 15
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Bass, H., K-Theory and stable algebra, Publ. Math. IHES 22 (1964), 5–70.CrossRefGoogle Scholar
2. Azumaya, G., Some characterizations of regular modules, Publ. Mat. 34 (1950), 241–248.Google Scholar
3. Camillo, V. P. and Khurana, D., A characterization of unit-regular rings, Comm. Algebra 29 (2001), 2293–2295.Google Scholar
4. Camillo, V. and Yu, H.-P., Stable range 1 for rings with many idempotents, Trans. Amer. Math. Soc. 347 (8) (1995), 3141–3147.Google Scholar
5. Chen, H. and Nicholson, W. K., Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra 218 (2014), 1431–1442.Google Scholar
6. Ehrlich, G., Units and one-sided units in regular rings, Trans. Amer. Math. Soc. 216 (1976), 81–90.CrossRefGoogle Scholar
7. Siddique, F., On two questions of Nicholson, preprint.Google Scholar
8. Khurana, D. and Lam, T. Y., Rings with internal cancellation, J. Algebra 284 (2005), 203–235.Google Scholar
9. Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278.CrossRefGoogle Scholar
10. Nicholson, W. K. and Sánchez Campos, E., Morphic modules, Comm. Algebra 33 (8) (2005), 2629–2647.Google Scholar
11. Vaserstein, L. N., Bass's first stable range condition, J. Pure. Appl. Algebra 34 (1984), 319–330.Google Scholar
12. Warfield, R. B., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 460–465.Google Scholar
13. Zelmanowitz, J. M., Regular modules, Trans. Amer. Math. Soc. 163 (1972), 341–355.Google Scholar