Skip to main content Accessibility help


  • H. CHEN (a1), W. K. NICHOLSON (a2) and Y. ZHOU (a3)


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.



Hide All
1. Bass, H., K-Theory and stable algebra, Publ. Math. IHES 22 (1964), 570.
2. Azumaya, G., Some characterizations of regular modules, Publ. Mat. 34 (1950), 241248.
3. Camillo, V. P. and Khurana, D., A characterization of unit-regular rings, Comm. Algebra 29 (2001), 22932295.
4. Camillo, V. and Yu, H.-P., Stable range 1 for rings with many idempotents, Trans. Amer. Math. Soc. 347 (8) (1995), 31413147.
5. Chen, H. and Nicholson, W. K., Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra 218 (2014), 14311442.
6. Ehrlich, G., Units and one-sided units in regular rings, Trans. Amer. Math. Soc. 216 (1976), 8190.
7. Siddique, F., On two questions of Nicholson, preprint.
8. Khurana, D. and Lam, T. Y., Rings with internal cancellation, J. Algebra 284 (2005), 203235.
9. Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269278.
10. Nicholson, W. K. and Sánchez Campos, E., Morphic modules, Comm. Algebra 33 (8) (2005), 26292647.
11. Vaserstein, L. N., Bass's first stable range condition, J. Pure. Appl. Algebra 34 (1984), 319330.
12. Warfield, R. B., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 460465.
13. Zelmanowitz, J. M., Regular modules, Trans. Amer. Math. Soc. 163 (1972), 341355.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed