In
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
, 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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
.
*Let ''F'' be a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 representa ...
of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
plays an important role in the
representation theory of Lie algebras
In the mathematics, mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrix (mathematics), matrices (or endomorphisms of a vector space) in s ...
. For example, a
Casimir element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ...
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
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simp ...
*
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 modul ...
Notes
References
*Bourbaki, ''Algebra''.
* Richard S. Pierce.
Associative algebras'. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982,
Ring theory
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