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A CLASS OF EXCHANGE RINGS

Published online by Cambridge University Press:  01 September 2008

TSIU-KWEN LEE  and
YIQIANG ZHOU
TSIU-KWEN LEE
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan. Member of Mathematics Division (Taipei Office), National Center for Theoretical Sciences e-mail: tklee@math.ntu.edu.tw
YIQIANG ZHOU*
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland A1C 5S7, Canada e-mail: zhou@math.mun.ca
*
*Corresponding author.
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Abstract

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It is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

Keywords

Primary 16U99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 509 - 522
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

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