# Center (algebra)

# 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.

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 (or an associative algebra)

*R*is the subset of*R*consisting of all those elements*x*of*R*such that*xr*=*rx*for all*r*in*R*.^{[3]}The center is a commutative subring of*R*.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*.

See also

Centralizer and normalizer

Center (category theory)

## References

*Monoids, Acts and Categories*. De Gruyter Expositions in Mathematics.

**29**. Walter de Gruyter. p. 25. ISBN 978-3-11-015248-7.

*Semigroups*. Translations of Mathematical Monographs.

**3**. Translated by A. A. Brown; J. M. Danskin; D. Foley; S. H. Gould; E. Hewitt; S. A. Walker; J. A. Zilber. Providence, Rhode Island: American Mathematical Soc. p. 96. ISBN 978-0-8218-8641-0.

*Modern Algebra: An Introduction*(3rd ed.). John Wiley and Sons. p. 118. ISBN 0-471-51001-7. The

*center*of a ring

*R*is defined to be {

*c*∈

*R*:

*cr*=

*rc*for every

*r*∈

*R*}., Exercise 22.22