Very clean matrices over local rings
  An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and sufficient conditions under which a triangular $2\times 2$ matrix ring over local rings is very clean. The very clean $2\times 2$ matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained. 
H. Chen, B. Ungor and S. Halicioglu
math.RA