turkmath.org

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.