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.
Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.
Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring.
Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
Examples
Some examples of noncommutative rings: The matrix ring of n-by-n matrices over the real numbers, where Hamilton's quaternions Any group ring constructed from a group that is not abelian
Some examples of rings that are not typically commutative (but may be commutative in simple cases): The free ring generated by a finite set, an example of two non-equal elements being The Weyl algebra , being the ring of polynomial differential operators defined over affine space; for example, , where the ideal corresponds to the commutator The quotient ring , called a quantum plane, where Any Clifford algebra can be described explicitly using an algebra presentation: given an -vector space of dimension with a quadratic form , the associated Clifford algebra has the presentation for any basis of , Superalgebras are another example of noncommutative rings; they can be presented as There are finite noncommutative rings: for example, the -by- matrices over a finite field, for . The smallest noncommutative ring is the ring of the upper triangular matrices over the field with two elements; it has eight elements and all noncommutative rings with eight elements are isomorphic to it or to its opposite.
History
Beginning with division rings arising from geometry, the study of noncommutative rings 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, Ø. Ore, J. Wedderburn and others.
Differences between commutative and noncommutative algebra
Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are 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 separately 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.
Important classes
Division rings
A division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with . Stated differently, a ring is a division ring if and only if its group of units is the set of all nonzero elements.
Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".
Semisimple rings
A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.