# Very clean matrices over local rings

 Received: 16.I.2015, Accepted: 29.XII.2015 Abstract. 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 in $R$. 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. Keywords: very clean ring, very clean matrix, local ring Mathematics Subject Classification (2010): 15A13, 15B99, 16L99 Authors: Burcu Ungor, Department of Mathematics, Ankara University, 06100 Ankara, Turkey Huanyin Chen, Department of Mathematics, Hangzhou Normal University, Hangzhou, China Sait Halicioglu, Department of Mathematics, Ankara University, 06100 Ankara, Turkey