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Concept
Von Neumann regular ring
Summary
In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of the element a; in general x is not uniquely determined by a. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left R-module is flat.
Von Neumann regular rings were introduced by under the name of "regular rings", in the course of his study of von Neumann algebras and continuous geometry. Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra.
An element a of a ring is called a von Neumann regular element if there exists an x such that a = axa. An ideal is called a (von Neumann) regular ideal if for every element a in there exists an element x in such that a = axa.
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. Every direct product of von Neumann regular rings is again von Neumann regular.
Another important class of examples of von Neumann regular rings are the rings Mn(K) of n-by-n square matrices with entries from some field K. If r is the rank of A ∈ Mn(K), Gaussian elimination gives invertible matrices U and V such that
(where Ir is the r-by-r identity matrix). If we set X = V−1U−1, then
More generally, the n × n matrix ring over any von Neumann regular ring is again von Neumann regular.
If V is a vector space over a field (or skew field) K, then the endomorphism ring EndK(V) is von Neumann regular, even if V is not finite-dimensional.
Generalizing the above examples, 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.
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