Almost Abelian rings

summary:A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N(R)$ and $e\in E(R)$, where $E(R)$ and $N(R)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N(R)$ is an ideal of $R$ and $R/N(R)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly regular if and only if $R$ is regular and left almost Abelian. (3) A left almost Abelian clean ring is an exchange ring. (4) For a left almost Abelian ring $R$, it is an exchange $(S,2)$ ring if and only if $\mathbb Z/2\mathbb Z$ is not a homomorphic image of $R$.