• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.125-134
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• 2013

An element in a ring R is strongly clean provided that it is the sum of an idempotent and a unit that commutate. In this note, several necessary and sufficient conditions under which a $2{\times}2$ matrix over an integral domain is strongly clean are given. These show that strong cleanness over integral domains can be characterized by quadratic and Diophantine equations.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.115-125
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• 2018

In this paper, we introduce the notion of a clean ordered Krasner hyperring and investigate some properties of it. Now, let (R, +, ${\cdot}$, ${\leq}$) be a clean ordered Krasner hyperring. The following is a natural question to ask: Is there a strongly regular relation ${\sigma}$ on R for which $R/{\sigma}$ is a clean ordered ring? Our motivation to write the present paper is reply to the above question.