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