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 general "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:(where "I"_"r" is the "r"-by-"r" identity matrix). If we set "X" = "V" ^-1"U" ^-1, then:

The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular.

A Boolean ring is a ring in which every element satisfies "a"² = "a". Every Boolean ring is von Neumann regular.

Facts

The following statements are equivalent for the ring "R":
* "R" is von Neumann regular
* every principal left ideal is generated by an idempotent
* every finitely generated left ideal is generated by an idempotent
* every principal left ideal is a direct summand of the left "R"-module "R"
* every finitely generated left ideal is a direct summand of the left "R"-module "R"
* every finitely generated submodule of a projective left "R"-module "P" is a direct summand of "P"
* every left "R"-module is flat: this is also known as "R" being absolutely flat, or "R" having weak dimension 0.
* every short exact sequence of left "R"-modules is pure exactThe corresponding statements for right modules are also equivalent to "R" being von Neumann regular.

In a commutative von Neumann regular ring, for each element "x" there is a unique element "y" such that "xyx"="x" and "yxy"="y", so there is a canonical way to choose the "weak inverse" of "x".The following statements are equivalent for the commutative ring "R":
* "R" is von Neumann regular
* "R" has Krull dimension 0 and no nonzero nilpotent elements
* Every localization of "R" at a maximal ideal is a field
*"R" is a subring of a product of fields closed under taking "weak inverses" of "x"∈"R" (the unique element "y" such that "xyx"="x" and "yxy"="y").

Every semisimple ring is von Neumann regular, and a 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 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.

Generalizations

A ring "R" is called "strongly von Neumann regular" if for every "a" in "R", there is some "x" in "R" with "a" = "aax". The condition is left-right symmetric. Every strongly von Neumann regular ring is semiprimitive, and even more, is a subdirect product of division rings. In some sense, this more closely mimics the properties of commutative von Neumann regular rings, which are subdirect products of fields. Of course for commutative rings, von Neumann regular and strongly von Neumann regular are equivalent. In general, the following are equivalent for a ring "R":
* "R" is strongly von Neumann regular
* "R" is von Neumann regular and reduced
* "R" is von Neumann regular and every idempotent in "R" is central
* Every principal left ideal of "R" is generated by a central idempotent

Further reading

* Ken Goodearl: "Von Neumann Regular Rings", 2nd ed. 1991
*springer|id=R/r080830|title=Regular ring (in the sense of von Neumann)|author=L.A. Skornyakov
*J. von Neumann, "Continuous geometries" , Princeton Univ. Press (1960)