Very cleanness of generalized matrices

Document Type: Research Paper

Author

Department of Mathematics‎, ‎Bilkent University, Ankara‎, ‎Turkey.

Abstract

An element $a$ in a ring $R$ is very clean in case there exists‎ ‎an idempotent $e\in R$ such that $ae = ea$ and either $a‎- ‎e$ or $a‎ + ‎e$ is invertible‎. ‎An element $a$ in a ring $R$ is very $J$-clean‎ ‎provided that there exists an idempotent $e\in R$ such that $ae =‎ ‎ea$ and either $a-e\in J(R)$ or $a‎ + ‎e\in J(R)$‎. ‎Let $R$ be a‎ ‎local ring‎, ‎and let $s\in C(R)$‎. ‎We prove that $A\in K_s(R)$ is‎ ‎very clean if and only if $A\in U(K_s(R))$‎, ‎$I\pm A\in U(K_s(R))$‎ ‎or $A\in K_s(R)$ is very J-clean‎.

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