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TWO QUESTIONS OF L. VAŠ ON
$\ast $-CLEAN RINGS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 08 March 2013, pp. 499-505
- Print publication:
- December 2013
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- Article
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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.
