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Concept
Noncommutative ring
Summary
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.
Some examples of noncommutative rings:
The matrix ring of n-by-n matrices over the real numbers, where n > 1
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 n 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 n-by-n matrices over a finite field, for n > 1. 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.
Official source
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