# Center (algebra)

The term

*center* is used in various contexts in

abstract algebra to denote the set of all those elements that

commute with all other elements. More specifically:

- 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*, so *R* is an associative 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*.
- 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 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*. See center of a group for more information.