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TWO QUESTIONS OF L. VAŠ ON $\ast $-CLEAN RINGS

Part of: Rings and algebras with additional structure Conditions on elements

Published online by Cambridge University Press:  08 March 2013

JIANLONG CHEN  and
JIAN CUI
JIANLONG CHEN
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, PR China email jlchen@seu.edu.cn
JIAN CUI*
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, PR China email jcui1635@gmail.com
*
✉jcui2006@126.com
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Abstract

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A $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

Keywords

∗-clean ringstrongly ∗-clean ring∗-regular ringstrongly clean ringunit regular ring

MSC classification

Secondary: 16U99: None of the above, but in this section 16W10: Rings with involution; Lie, Jordan and other nonassociative structures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 499 - 505
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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