In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, 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
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
*The center of the (full)
matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the
identity matrix.
*Let ''F'' be a
field extension of a field ''k'', and ''R'' an algebra over ''k''. Then
*The center of the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
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.
*The center of a
simple algebra is a field.
See also
*
Center of a group
*
Central simple algebra
*
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
Notes
References
*Bourbaki, ''Algebra''.
* Richard S. Pierce.
Associative algebras'. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982,
Ring theory
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