von Neumann regular (original) (raw)

An element a of a ring R is said to be von Neumann regular if there exists b∈R such that a⁢b⁢a=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 M2⁢(ℝ), the ring of 2×2 matrices over ℝ. Then

(2000)=(2000)⁢(12000)⁢(2000)

is von Neumann regular. In fact, we can replace 2 with any non-zero r∈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 r⁢s=f with e⁢f=e. Then r⁢e is von Neumann regular, with s,s⁢e and s⁢f all as pseudoinverses. In another generalization, we have two rings R,S where R is an algebra over S. Take any idempotent e∈R, and any invertible element s∈S such that s commutes with e. Then s⁢e 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.