# QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS

• Cui, Jian (Department of Mathematics Anhui Normal University) ;
• Yin, Xiaobin (Department of Mathematics Anhui Normal University)
• Published : 2014.05.31
• 99 6

#### Abstract

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.

#### Keywords

quasipolar ring;matrix ring;strongly clean ring;local ring

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