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On two open problems about strongly clean rings

Published online by Cambridge University Press:  17 April 2009

Zhou Wang  and
Jianlong Chen
Zhou Wang
Affiliation:
Department of Mathematics, Southeast University, Nanjing, 210096, Peoples Republic of China e-mail: fylwangz@163.com, jlchen@seu.edu.cn
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A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 279 - 282
Copyright
Copyright © Australian Mathematical Society 2004

References

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