# 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
; "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 - 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.

## Notes

- "vector spaces - A linear operator commuting with all such operators is a scalar multiple of the identity. - Mathematics Stack Exchange".
*Math.stackexchange.com*. Retrieved 2017-07-22.

## 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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