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# von Neumann regular

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.

## Mathematics Subject Classification

16E50*no label found*

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