Center (ring theory)

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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)displaystyle 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.

Contents

• 
  1 Examples
  


• 
  2 See also
  


• 
  3 Notes
  


• 
  4 References
  

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 Z(R⊗kF)=Z(R)⊗kF.displaystyle Z(Rotimes _kF)=Z(R)otimes _kF.
  


• 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

Notes

1. 
   ^ "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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