# Center (algebra)

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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 a natural isomorphism satisfying certain axioms.

## See also

## References

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