High Quality Content by WIKIPEDIA articles! 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 general x is not uniquely determined by a. 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 semisimple ring is von Neumann regular, and a left (or right) Noetherian von Neumann regular ring is semisimple. Every von Neumann regular ring has Jacobson radical {0} and is thus semiprimitive (also called "Jacobson semi-simple"). 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.

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