Center (ring theory)

In algebra, the center of a ring Template:Mvar is the subring consisting of the elements Template:Mvar such that Template:Nowrap for all elements Template:Mvar in Template:Mvar. It is a commutative ring and is denoted as Template:Math; 'Z' stands for the German word Template:Lang, meaning "center".

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

Examples

• The center of a commutative ring Template:Mvar is Template:Mvar itself.
• The center of a skew-field is a field.
• The center of the (full) matrix ring with entries in a commutative ring Template:Mvar consists of Template:Mvar-scalar multiples of the identity matrix.
• Let Template:Mvar be a field extension of a field Template:Mvar, and Template:Mvar an algebra over Template:Mvar. Then $\mathrm{Z}(R{\otimes }_{k}F)=\mathrm{Z}(R){\otimes }_{k}F$.
• 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

References

• Template:Cite book
• Template:Cite book

Template:Algebra-stub