# 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 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 is clear from context) if every element of $R$ is von Neumann regular.

Title von Neumann regular VonNeumannRegular 2013-03-22 12:56:18 2013-03-22 12:56:18 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 16E50 von Neumann regular ring regular ring pseudoinverse