In algebra, the center of a ringR 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.
^"vector spaces - A linear operator commuting with all such operators is a scalar multiple of the identity. - Mathematics Stack Exchange". Math.stackexchange.com. Retrieved July 22, 2017.
ReferencesEdit
Bourbaki, Algebra.
Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, ISBN978-0-387-90693-5
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center, ring, theory, algebra, center, ring, subring, consisting, elements, such, that, elements, commutative, ring, denoted, displaystyle, stands, german, word, zentrum, meaning, center, ring, then, associative, algebra, over, center, conversely, associative,. 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 ReferencesExamples EditThe 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 k F Z R k F displaystyle Z left R otimes k F right Z R otimes k F nbsp 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 EditCenter of a group Central simple algebra Morita equivalenceNotes Edit vector spaces A linear operator commuting with all such operators is a scalar multiple of the identity Mathematics Stack Exchange Math stackexchange com Retrieved July 22 2017 References EditBourbaki Algebra Richard S Pierce Associative algebras Graduate texts in mathematics Vol 88 Springer Verlag 1982 ISBN 978 0 387 90693 5 nbsp This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Center ring theory amp oldid 1147730941, wikipedia, wiki, book, books, library,