Noncommutative ring
Encyclopedia
In mathematics
, more specifically modern algebra and ring theory
, a noncommutative ring is a ring
whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.
Noncommutative rings are ubiquitous in mathematics, and occur in numerous sciences. For instance, matrix multiplication
is never commutative, except in trivial cases, despite the fact that matrices arise naturally as rings of linear transformation
s of some vector space over a field. Furthermore, mathematical physics
and more generally linear algebra
exploit the concept of a matrix often. Noncommutative rings also arise naturally in the representation theory
of groups. Algebras
, and more specifically group algebras
, occur also in noncommutative ring theory.
The study of noncommutative rings is a major area of modern algebra. Influential work by Richard Brauer
, Nathan Jacobson
, I. N. Herstein and P. M. Cohn and other mathematicians, has led to much of modern day ring theory. Basic but influential concepts in the field include the Jacobson radical
, the Jacobson density theorem
, the Artin–Wedderburn theorem
s and the Brauer group
.
s do not. As an example, there exist rings which contain non-trivial proper left or right ideals
, but are still simple
; that is contain no non-trivial proper (two-sided) ideals.
The theory of vector space
s is one illustration of a special case of an object studied in noncommutative ring theory. In linear algebra
, the "scalars of a vector space" are required to lie in a field
, that is, a commutative division ring
. The concept of a module
, however, requires only that the scalars lie in an abstract ring. Neither commutativity nor the division ring assumption is required on the scalars in this case. Module theory has various applications in noncommutative ring theory, as one can often obtain information about the structure of a ring by making use of its modules. The concept of the Jacobson radical
of a ring; that is, the intersection of all right/left annihilators
of simple
right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also remarkable that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative. Therefore, the Jacobson radical also captures a concept which may seem to be not well-defined for noncommutative rings.
Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of n by n matrices over a field
is noncommutative despite its natural occurrence in physics
. More generally, endomorphism ring
s of abelian groups are rarely commutative.
Noncommutative rings, like noncommutative groups, are not very well understood. For instance, although every finite abelian group
is the direct sum of (finite) cyclic group
s of prime-power order, non-abelian groups do not possess such a simple structure. Likewise, various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the nilradical
, although "innocent" in nature, need not be an ideal unless the ring is assumed to be commutative. Specifically, the set of all nilpotent elements in the ring of all n x n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. Therefore, the notion of the nilradical, as it stands, cannot be studied in noncommutative ring theory. Note however that there are analogues of the nilradical defined for noncommutative rings, that coincide with the notion of the nilradical when commutativity is assumed.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically modern algebra and ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a noncommutative ring is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.
Noncommutative rings are ubiquitous in mathematics, and occur in numerous sciences. For instance, matrix multiplication
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is never commutative, except in trivial cases, despite the fact that matrices arise naturally as rings of linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s of some vector space over a field. Furthermore, mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
and more generally linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
exploit the concept of a matrix often. Noncommutative rings also arise naturally in the representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of groups. Algebras
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
, and more specifically group algebras
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
, occur also in noncommutative ring theory.
The study of noncommutative rings is a major area of modern algebra. Influential work by Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
, Nathan Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
, I. N. Herstein and P. M. Cohn and other mathematicians, has led to much of modern day ring theory. Basic but influential concepts in the field include the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
, the Jacobson density theorem
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R....
, the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
s and the Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
.
Discussion
Often noncommutative rings possess interesting invariants that commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s do not. As an example, there exist rings which contain non-trivial proper left or right ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
, but are still simple
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
; that is contain no non-trivial proper (two-sided) ideals.
The theory of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s is one illustration of a special case of an object studied in noncommutative ring theory. In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the "scalars of a vector space" are required to lie in a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, that is, a commutative division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
. The concept of a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, however, requires only that the scalars lie in an abstract ring. Neither commutativity nor the division ring assumption is required on the scalars in this case. Module theory has various applications in noncommutative ring theory, as one can often obtain information about the structure of a ring by making use of its modules. The concept of the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
of a ring; that is, the intersection of all right/left annihilators
Annihilator (ring theory)
In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
of simple
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also remarkable that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative. Therefore, the Jacobson radical also captures a concept which may seem to be not well-defined for noncommutative rings.
Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of n by n matrices over a field
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is noncommutative despite its natural occurrence in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. More generally, endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
s of abelian groups are rarely commutative.
Noncommutative rings, like noncommutative groups, are not very well understood. For instance, although every finite abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is the direct sum of (finite) cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s of prime-power order, non-abelian groups do not possess such a simple structure. Likewise, various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the nilradical
Nilradical
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring. In the non-commutative ring case, more care is needed resulting in several related radicals.- Commutative rings :...
, although "innocent" in nature, need not be an ideal unless the ring is assumed to be commutative. Specifically, the set of all nilpotent elements in the ring of all n x n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. Therefore, the notion of the nilradical, as it stands, cannot be studied in noncommutative ring theory. Note however that there are analogues of the nilradical defined for noncommutative rings, that coincide with the notion of the nilradical when commutativity is assumed.