von neumann regular ring

In mathematics, a ring R is von Neumann regular if for every a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of a; note however that in most cases x is not uniquely determined by a. (The regular rings of commutative algebra are unrelated.)

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 M_n(K) of n-by-n square matrices with entries from some field K. If r is the rank of A∈M_n(K), then there exist invertible matrices U and V such that

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

0 &0\end{pmatrix} V (where I_r 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

Facts

While the above definition of von Neumann regularity seems somewhat contrived and technical, there are several conceptual reformulations. The following statements are equivalent for the ring R:

R is von Neumann regular
every finitely generated submodule of a projective left R module P is a direct summand of P
every left R-module is flat
every short exact sequence of left R-modules is pure exact

The corresponding statements for right modules are also equivalent to R being von Neumann regular. Every von Neumann regular ring has Jacobson radical {0} and is thus semiprimitive. Generalizing the above example, suppose S is some ring and M is an S-module such that every submodule of M is a direct summand of M (such modules M are called semisimple). Then the endomorphism ring End_S(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular. A ring is semisimple artinian if and only if it is von Neumann regular and left (or right) Noetherian.

Von Neumann Regular Ring

Examples

Facts

Further reading