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 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. Let F be a field extension of a field k, and R an algebra over k. Then Z ( R ⊗ k F ) = Z ( R ) ⊗ k F {\displaystyle \operatorname {Z} (R\otimes _{k}F)=\operatorname {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
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