Center (ring theory)
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 Zentrum, 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 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.