Von Neumann regular ring

In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R such that a = axa. To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat.

One may think of x as a "weak inverse" of a. In general x is not uniquely determined by a.

Von Neumann regular rings were introduced by von Neumann (1936) under the name of "regular rings", during his study of von Neumann algebras and continuous geometry.

An element a of a ring is called a von Neumann regular element if there exists an x such that a = axa.^[1] An ideal ${\mathfrak {i}}$ is called a (von Neumann) regular ideal if it is a von Neumann regular non-unital ring, i.e. if for every element a in ${\mathfrak {i}}$ there exists an element x in ${\mathfrak {i}}$ such that a = axa.^[2]

Examples

Every field (and every skew field) is von Neumann regular: for a ≠ 0 we can take x = a⁻¹.^[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

A=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V

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

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.

More generally, the matrix ring over a von Neumann regular ring is again a von Neumann regular ring.^[1]

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 a² = 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 element
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 exact

The 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 is reduced
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).

Also, the following are equivalent: for a commutative ring A

R = A / nil(A) is von Neumann regular.
The spectrum of A is Hausdorff (in the Zariski topology).
The constructible topology and Zariski topology for Spec(A) coincide.

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 End_S(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular.

Generalizations and specializations

Special types of von Neumann regular rings include unit regular rings and strongly von Neumann regular rings and rank rings.

A ring R is called unit regular if for every a in R, there is a unit u in R such that a = aua. Every semisimple ring is unit regular, and unit regular rings are directly finite rings. An ordinary von Neumann regular ring need not be directly finite.

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. Strongly von Neumann regular rings are unit regular. Every strongly von Neumann regular ring 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

Generalizations of von Neumann regular rings include π-regular rings, left/right semihereditary rings, left/right nonsingular rings and semiprimitive rings.

References

1 2 3 Kaplansky (1972) p.110
↑ Kaplansky (1972) p.112

Kaplansky, Irving (1972), Fields and rings, Chicago lectures in mathematics (Second ed.), University of Chicago Press, ISBN 0-226-42451-0, Zbl 1001.16500

Von Neumann regular ring

Examples

Facts

Generalizations and specializations

See also

References

Further reading