# Centralizer and normalizer

Suitably formulated, the definitions also apply to monoids and semigroups.

In ring theory, the **centralizer of a subset of a ring** is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring *R* is a subring of *R*. This article also deals with centralizers and normalizers in a Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

The **centralizer** of a subset *S* of group (or semigroup) *G* is defined as^{[3]}

where again only the first definition applies to semigroups. The definitions are similar but not identical. If *g* is in the centralizer of *S* and *s* is in *S*, then it must be that *gs* = *sg*, but if *g* is in the normalizer, then *gs* = *tg* for some *t* in *S*, with *t* possibly different from *s*. That is, elements of the centralizer of *S* must commute pointwise with *S*, but elements of the normalizer of *S* need only commute with *S as a set*. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure.

If *R* is a ring or an algebra over a field, and *S* is a subset of *R*, then the centralizer of *S* is exactly as defined for groups, with *R* in the place of *G*.

The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If *R* is an associative ring, then *R* can be given the bracket product [*x*, *y*] = *xy* − *yx*. Of course then *xy* = *yx* if and only if [*x*, *y*] = 0. If we denote the set *R* with the bracket product as L_{R}, then clearly the *ring centralizer* of *S* in *R* is equal to the *Lie ring centralizer* of *S* in L_{R}.