Von Neumann Regular Ring - Examples

Examples

Every field (and every skew field) is von Neumann regular: for a≠0 we can take x = a -1. An integral domain is von Neumann regular if and only if it is a field.

Another example of a von Neumann regular ring is the ring Mn(K) of n-by-n square matrices with entries from some field K. If r is the rank of A∈Mn(K), then there exist invertible matrices U and V such that

A = U \begin{pmatrix}I_r &0\\ 0 &0\end{pmatrix} V

(where Ir is the r-by-r identity matrix). If we set X = V -1U -1, then

AXA= U \begin{pmatrix}I_r &0\\ 0 &0\end{pmatrix} \begin{pmatrix}I_r &0\\ 0 &0\end{pmatrix} V = U \begin{pmatrix}I_r &0\\ 0 &0\end{pmatrix} V = A.

More generally, the matrix ring over a von Neumann regular ring is again a von Neumann regular ring.

The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular.

A Boolean ring is a ring in which every element satisfies a2 = a. Every Boolean ring is von Neumann regular.

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