DECOMPOSITIONS OF $2\times 2$ MATRICES OVER LOCAL RINGS

Huanyin Chen, Sait Halicioglu, and Handan Kose

Abstract: An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e{\in }^2\left(a\right)$ such that $a-e\in U\left(R\right)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we completely determine when every $2\times 2$ matrix and triangular matrix over local rings are perfectly clean. These give more explicit characterizations of strongly clean matrices over local rings. We also obtain several criteria for a triangular matrix to be perfectly J-clean. For instance, it is proved that for a commutative local ring $R$, every triangular matrix is perfectly J-clean in $T_n\left(R\right)$ if and only if $R$ is strongly J-clean.

Keywords: perfectly clean ring; perfectly J-clean ring; quasipolar ring; matrix; triangular matrix

Classification (MSC2000): 16S50; 16S70; 16U99

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