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A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N\left(R\right)$ and $e\in E\left(R\right)$, where $E\left(R\right)$ and $N\left(R\right)$ 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 $N\left(R\right)$ is an ideal of $R$ and $R/N\left(R\right)$ 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...

We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.