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RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT

Published online by Cambridge University Press:  19 May 2004

W. K. NICHOLSON  and
Y. ZHOU
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 A1C 5S7, Canada e-mail: [email protected]
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Abstract

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An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

Keywords

Primary 16E50secondary 16U9916S70
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 46 , Issue 2 , May 2004 , pp. 227 - 236
Copyright
2004 Glasgow Mathematical Journal Trust