Center (ring theory)

In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and is denoted as ${\displaystyle Z(R)}$; "Z" stands for the German word Zentrum, meaning "center".

If R is a ring, then R is an associative algebra over its center. Conversely, if R is an associative algebra over a commutative subring S, then S is a subring of the center of R, and if S happens to be the center of R, then the algebra R is called a central algebra.

Examples[]

• The center of a commutative ring R is R itself.
• The center of a skew-field is a field.
• The center of the (full) matrix ring with entries in a commutative ring R consists of R-scalar multiples of the identity matrix.^[1]
• Let F be a field extension of a field k, and R an algebra over k. Then ${\displaystyle Z\left(R\otimes _{k}F\right)=Z(R)\otimes _{k}F.}$
• The center of the universal enveloping algebra of a Lie algebra plays an important role in the representation theory of Lie algebras. For example, a Casimir element is an element of such a center that is used to analyze Lie algebra representations.
  See also: Harish-Chandra isomorphism
• The center of a simple algebra is a field.

See also[]

• Center of a group
• Central simple algebra
• Morita equivalence

Notes[]

References[]

• Bourbaki, Algebra.
• Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN 978-0-387-90693-5

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