Von Neumann regular ring

Last updated December 08, 2024

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 von Neumann ( 1936 ) 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.^[1] An ideal ${\mathfrak {i}}$ is called a (von Neumann) regular ideal 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. 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 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), Gaussian elimination gives 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 n × n matrix ring over any von Neumann regular ring is again von Neumann regular.^[1]

If V is a vector space over a field (or skew field) K, then the endomorphism ring End_K(V) is von Neumann regular, even if V is not finite-dimensional.^[3]

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 End_S(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular. Indeed, the semisimple rings are precisely the Noetherian von Neumann regular rings.

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.

Every von Neumann regular ring has Jacobson radical {0} and is thus semiprimitive (also called "Jacobson semi-simple").

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).
R is a V-ring.^[4]
R has the right-lifting property against the ring homomorphism Z[t] → Z[t^±] × Z determined by t ↦ (t, 0), or said geometrically, every regular function ${\textstyle \mathrm {Spec} (R)\to \mathbb {A} ^{1}}$ factors through the morphism of schemes $\{0\}\sqcup \mathbb {G} _{m}\to \mathbb {A} ^{1}$ .^[5]

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.

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. 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.

Notes

Related Research Articles

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In abstract algebra, an element $a$ of a ring $R$ is called a left zero divisor if there exists a nonzero $x$ in $R$ such that $ax = 0$ , or equivalently if the map from $R$ to $R$ that sends $x$ to $ax$ is not injective. Similarly, an element $a$ of a ring is called a right zero divisor if there exists a nonzero $y$ in $R$ such that $ya = 0$ . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element $a$ that is both a left and a right zero divisor is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same.

<span class="mw-page-title-main">Jacobson radical</span> Structure in Ring Theory (Mathematics)

In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Jacobson 1945.

In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map $as additive identity and the identity map as multiplicative identity.$

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.

In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element $a$ such that $a 2 = a$ . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that $a = a 2 = a 3 = a 4 = ... = a n$ for any positive integer $n$ . For example, an idempotent element of a matrix ring is precisely an idempotent matrix.

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a matrix ring denoted M_n(R) (alternative notations: Mat_n(R) and R^n×n). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs.

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent if their categories of modules are additively equivalent. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary.

In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.

In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annihilators of various sets.

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.

References

Burklund, Robert; Schlank, Tomer M.; Yuan, Allen (2022-07-20). "The Chromatic Nullstellensatz". p. 50. arXiv: 2207.09929 [math.AT].
Kaplansky, Irving (1972), Fields and rings, Chicago lectures in mathematics (Second ed.), University of Chicago Press, ISBN 0-226-42451-0, Zbl 1001.16500
Michler, G.O.; Villamayor, O.E. (April 1973). "On rings whose simple modules are injective". Journal of Algebra . 25 (1): 185–201. doi: 10.1016/0021-8693(73)90088-4 . hdl: 20.500.12110/paper_00218693_v25_n1_p185_Michler .
Skornyakov, L.A. (2001) [1994], "Regular ring (in the sense of von Neumann)", Encyclopedia of Mathematics , EMS Press
von Neumann, John (1936), "On Regular Rings", Proc. Natl. Acad. Sci. USA, 22 (12): 707–713, Bibcode:1936PNAS...22..707V, doi: 10.1073/pnas.22.12.707 , JFM 62.1103.03, PMC 1076849 , PMID 16577757, Zbl 0015.38802