von Neumann regular ring

Remarks

If $a x a = a$, then $e = a x$ is an idempotent with $e R = a R$; so $R$ is von Neumann regular if and only if every principal right ideal is generated by an idempotent. A commutative ring is von Neumann regular if and only if it is reduced of Krull dimension $0$, and if and only if every $R$-module is flat. Every semisimple ring, for instance the ring of $n \times n$ matrices over a division ring, is von Neumann regular.

Example: a Boolean ring $R$, in which $a^2 = a$ for all $a$, is von Neumann regular with $x = a$: $a a a = a^2 = a$. Example: a field is von Neumann regular, with $x = a^{-1}$ for $a \ne 0$ and $x = 0$ for $a = 0$; so is any product of fields, componentwise.

Non-example: $\Z$ is not von Neumann regular. For $a = 2$, the condition $2 x 2 = 2$ means $4x = 1$, which has no solution in $\Z$.