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
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