Noncommutative ring

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 transformations 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 theorems and the Brauer group.

Discussion

Often noncommutative rings possess interesting invariants that commutative rings 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 spaces 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 rings 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 groups 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.

References

• I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2. 
• I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America. ISBN 0-88385-015-X. 

See also

• Representation theory (group theory)