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PRODUCTS OF IDEMPOTENT ENDOMORPHISMS OF RELATIVELY FREE ALGEBRAS WITH WEAK EXCHANGE PROPERTIES

Published online by Cambridge University Press:  17 May 2007

John Fountain  and
Victoria Gould
John Fountain
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK ([email protected]; [email protected])
Victoria Gould
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK ([email protected]; [email protected])
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Abstract

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If $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

Keywords

Primary 20M2008A35Secondary 20M1008A05basisexchange propertyendomorphism monoididempotents
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 50 , Issue 2 , June 2007 , pp. 343 - 362
Copyright
Copyright © Edinburgh Mathematical Society 2007