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EHRLICH’S THEOREM FOR GROUPS

Published online by Cambridge University Press:  13 January 2010

YUANLIN LI
Affiliation:
Department of Mathematics, Brock University, St Catharines, Ontario, Canada L2S 3A1 (email: Yuanlin.Li@brocku.ca)
W. K. NICHOLSON
Affiliation:
Department of Mathematics, University of Calgary, Calgary, Canada T2N 1N4 (email: wknichol@ucalgary.ca)
Corresponding
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Abstract

A group G is called morphic if every endomorphism α:GG for which G satisfies G/≅ker (α). Call an endomorphism α∈end(G) regular if αβα=α for some β∈end(G), and call α unit regular if β can be chosen to be an automorphism of G. The main purpose of this paper is to prove the following group-theoretic analogue of a theorem of Ehrlich: if G is a morphic group, an endomorphism α:GG for which G is unit regular if and only if it is regular. As an application, a cancellation theorem is proved that characterizes the morphic groups among those with regular endomorphism monoids.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research was supported in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.

References

[1]Azumaya, G., ‘On generalized semi-primary rings and Krull–Remak–Schmidt’s theorem’, Japan. J. Math. 29 (1948), 525547.Google Scholar
[2]Ehrlich, G., ‘Units and one-sided units in regular rings’, Trans. Amer. Math. Soc. 216 (1976), 8190.CrossRefGoogle Scholar
[3]Li, Y., Nicholson, W. K. and Zan, L., Morphic groups, to appear.Google Scholar
[4]Nicholson, W. K. and Sánchez Campos, E., ‘Morphic modules’, Comm. Algebra 33 (2005), 26292647.CrossRefGoogle Scholar
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