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·b ≠ b·a. The term "noncommutative" means "not necessarily commutative" and thus "noncommutative" results apply to commutative rings as a subclass.
As mentioned, any commutative ring qualifies as a noncommutative ring, but some examples of rings which are not commutative follow:
- the matrix ring of n-by-n matrices over the real numbers, where n>1.
- Hamilton's quaternions.
- any group algebra made from a group that is not abelian.
Beginning with division rings arising from geometry, the study of noncommutative rings is has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes E. Artin, Richard Brauer, P. M. Cohn, W. R. Hamilton, I. N. Herstein, N. Jacobson, K. Morita, E. Noether, O. Ore and others.
Because noncommutative rings are a much larger class of rings than the subclass of commutative rings, their structure and behavior is less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutative is the necessity to consider right ideals and left ideals. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left-right distinction does not exist.
Classical results on noncommutative rings include:
- study of division rings, semisimple rings, semiprimitive rings, simple rings, and rings whose left ideals (or right ideals) satisfy some condition.
- structure theorems like Wedderburn's little theorem and its generalizations: the Artin-Wedderburn theorem and the Jacobson density theorem.
- generalizations of important commutative concepts such as Nakayama's lemma and noncommutative localization.
- categorical results such as the Morita theorems and derived concepts of Morita equivalence and the Brauer group.
- the Ore conditions on noncommutative domains which characterize when the noncommutative analogue of the field of fractions is possible, as well as generalizations such as Goldie's theorem.
- Noncommutative harmonic analysis
- Representation theory (group theory)
- Derived algebraic geometry
- Noncommutative algebraic geometry