algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the center of a ring ''R'' is the

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...

and is denoted as Z(''R''); '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'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

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 (or, occasionally, a sfield), 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 ...

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 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'') (alternat ...

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. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

. * Let ''F'' be a

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' 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 ident ...

plays an important role in the

representation theory of Lie algebras 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 ...

. 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 opera ...

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 i ...

s. See also: ''

Harish-Chandra isomorphism In mathematics, the Harish-Chandra isomorphism, introduced by , is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathf ...

''. * 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 sim ...

is a field.

Notes

References

* * {{algebra-stub Ring theory