Noncommutative ring

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In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not necessarily commutative; that is, there may exist a and b in R with a·bb·a.

The term "noncommutative" is sometimes used interchangeably with "not commutative." However, properly speaking, the term is meant to emphasize that commutativity is not assumed, but may be present. "Noncommutative" results apply equally well to commutative rings as a subclass.

Noncommutative rings are ubiquitous in mathematics, and occur in numerous sciences. For instance, matrix multiplication which arises naturally in rings of linear transformations of vector spaces over a field, is never commutative except in the case of 1-by-1 matrices. Such rings are the main object of study in linear algebra. 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 theorems and the Brauer group.

Discussion[edit]

Noncommutative rings can exhibit interesting features that commutative rings do not. For instance, there exist rings which have non-trivial proper left or right ideals, but which lack non-trivial (two-sided) ideals: these are called simple rings. The 2-by-2 matrix ring over a field is an example of such a ring. A particular right ideal is given by the subset of matrices with zeros in the bottom row, and a particular left ideal is given by the subset of matrices whose right column is all zeros.

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 rings of abelian groups are rarely commutative.

The theory of vector spaces is one illustration 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 data which may seem to be not well-defined for noncommutative rings.

The structure of noncommutative rings is less well understood than commutative rings, as non-abelian groups are less well understood than abelian groups. For instance, although every finite abelian group is the direct sum of (finite) cyclic groups of prime-power order, non-abelian groups do not possess such a simple structure. Likewise, familiar theorems about commutative rings may or may not have analogues in the general noncommutative case. The definition of the nilradical of a commutative ring, for example, cannot be directly applied to noncommutative rings, since the set of nilpotent elements need not be an ideal. The example of the ring of all n x n matrices over a field always has nonzero nilpotent elements, but the set never forms an ideal, since the ring is simple. Therefore, the notion of the nilradical, as it stands, cannot be studied in noncommutative ring theory. There are generalizations of the nilradical defined for noncommutative rings which coincide with the notion of the nilradical when commutativity is assumed.

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