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An element $a$ of a ring $R$ is said to be von Neumann regular if there exists $b\in R$ such that $aba=a$ . Such an element $b$ is known as a pseudoinverse of $a$ .
For example, any unit in a ring is von Neumann regular. Also, any idempotent element is von Neumann regular. For a non-unit, non-idempotent von Nuemann regular element, take $M_2(\mathbb{R})$ , the ring of $2\times 2$ matrices over $\mathbb{R}$ . Then
$\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}= \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$
is von Neumann regular. In fact, we can replace $2$ with any non-zero $r\in \mathbb{R}$ and the resulting matrix is also von Neumann regular. There are several ways to generalize this example. One way is take a central idempotent $e$ in any ring $R$ , and any $rs=f$ with $ef=e$ . Then $re$ is von Neumann regular, with $s,se$ and $sf$ all as pseudoinverses. In another generalization, we have two rings $R,S$ where $R$ is an algebra over $S$ . Take any idempotent $e\in R$ , and any invertible element $s\in S$ such that $s$ commutes with $e$ . Then $se$ is von Neumann regular.
A ring $R$ is said to be a von Neumann regular ring (or simply a regular ring, if the meaning is clear from context) if every element of $R$ is von Neumann regular.
For example, any division ring is von Neumann regular, and so is any ring of matrices over a division ring. In general, any semisimple ring is von Neumann regular.
Remark. Note that regular ring in the sense of von Neumann should not be confused with regular ring in the sense of commutative algebra, which is a Noetherian ring whose localization at every prime ideal is a regular local ring.
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