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EHRLICH’S THEOREM FOR GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 January 2010

YUANLIN LI  and
W. K. NICHOLSON
YUANLIN LI
Affiliation:
Department of Mathematics, Brock University, St Catharines, Ontario, Canada L2S 3A1 (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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A group G is called morphic if every endomorphism α:G→G for which Gα◃G satisfies G/Gα≅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:G→G for which Gα◃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.

Keywords

group endomorphismsunit regular endomorphismsEhrlich’s theorem

MSC classification

Secondary: 20F28: Automorphism groups of groups 20F06: Cancellation theory; application of van Kampen diagrams
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 304 - 309
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), 525–547.Google Scholar
[2]Ehrlich, G., ‘Units and one-sided units in regular rings’, Trans. Amer. Math. Soc. 216 (1976), 81–90.Google 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), 2629–2647.Google Scholar