# 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

$\left(\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$

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.

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 , which is a Noetherian ring^{} whose localization^{} at every prime ideal^{} is a regular local ring^{}.

Title | von Neumann regular |
---|---|

Canonical name | VonNeumannRegular |

Date of creation | 2013-03-22 12:56:18 |

Last modified on | 2013-03-22 12:56:18 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 13 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 16E50 |

Defines | von Neumann regular ring |

Defines | regular ring |

Defines | pseudoinverse |