In algebra, the center of a ring is the subring consisting of the elements such that for all elements in . It is a commutative ring and is denoted as ; 'Z' stands for the German word , meaning "center".
If is a ring, then is an associative algebra over its center. Conversely, if is an associative algebra over a commutative subring , then is a subring of the center of , and if happens to be the center of , then the algebra is called a central algebra.
Examples The center of a commutative ring is itself. The center of a skew-field is a field. The center of the (full) matrix ring with entries in a commutative ring consists of -scalar multiples of the identity matrix. Let be a field extension of a field , and an algebra over . Then . 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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