• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1069-1078
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• 2019

Let R be a ring with identity and P(R) denote the prime radical of R. An element r of a ring R is called strongly P-clean, if there exists an idempotent e such that $r-e=p{\in}P$(R) with ep = pe. In this paper, we determine necessary and sufficient conditions for an element of a trivial Morita context to be strongly P-clean.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.