In
algebra, the center of a ring ''R'' is the
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
consisting of the elements ''x'' such that ''xy = yx'' for all elements ''y'' in ''R''. It is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
and is denoted as
; "Z" stands for the German word ''Zentrum'', meaning "center".
If ''R'' is a ring, then ''R'' is an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
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
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
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 o ...
.
*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 of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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 representation
In the 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 matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...
s.
*The center of a
simple algebra In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simple ...
is a field.
See also
*
Center of a group
In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation,
:.
The center is a normal subgroup, . As a subg ...
*
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 simple a ...
*
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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