Von Neumann regular ring

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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 local 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

<math>A = U \begin{pmatrix}I_r &0\\

0 &0\end{pmatrix} V</math> (where I_r is the r-by-r identity matrix). If we set X = V ^-1U ^-1, then

<math>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</math>

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.

Further reading

• Ken Goodearl: Von Neumann Regular Rings, 2nd ed. 1991