Center (algebra)

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:

The center of a group G consists of all those elements x in G such that xg = gx for all g in G. This is a normal subgroup of G.
The similarly named notion for a semigroup is defined likewise and it is a subsemigroup.[1][2]
The center of a ring R is the subset of R consisting of all those elements x of R such that xr = rx for all r in R. The center is a commutative subring of R, and R is an algebra over its center.
The center of an algebra A consists of all those elements x of A such that xa = ax for all a in A. See also: central simple algebra.
The center of a Lie algebra L consists of all those elements x in L such that [x,a] = 0 for all a in L. This is an ideal of the Lie algebra L.
The center of a monoidal category C consists of pairs (A,u) where A is an object of C, and \( u:A \otimes - \rightarrow - \otimes \) A a natural isomorphism satisfying certain axioms.

See also

Centralizer and normalizer

References

Mati Kilp; M. Kilʹp; U. Knauer; Aleksandr Vasilʹevich Mikhalev (2000). Monoids, Acts, and Categories: With Applications to Wreath Products and Graphs : a Handbook for Students and Researchers. Walter de Gruyter. p. 25. ISBN 978-3-11-015248-7.

E. S. Li͡apin (1968). Semigroups. American Mathematical Soc. p. 96. ISBN 978-0-8218-8641-0.

Modern Algebra, R. Durbin, 3rd edition (1992), page 118, exercise 22.22

Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximal ideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomial p(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields Fp = Z / pZ.