Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T03:06:46.689Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Anderson, D.D. and Camillo, V.P., ‘Commutative rings whose elements are a sum of a unit and idempotent’, Comm. Algebra 30 (2002), 3327–3336.Google Scholar
[2]Camillo, V.P. and Yu, H.P., ‘Exchange rings, units and idempotents’, Comm. Algebra 22 (1994), 4737–4749.CrossRefGoogle Scholar
[3]Camillo, V.P. and Khurana, D., ‘A characterization of unit regular rings’, Comm. Algebra 29 (2001), 2293–2295.Google Scholar
[4]Han, J. and Nicholson, W.K., ‘Extensions of clean rings’, Comm. Algebra 29 (2001), 2589–2595.Google Scholar
[5]Nicholson, W.K., ‘Strongly clean rings and Fitting's lemma’, Comm. Algebra 27 (1999), 3583–3592.CrossRefGoogle Scholar