Centralizer and normalizer
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
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 LR, then clearly the ring centralizer of S in R is equal to the Lie ring centralizer of S in LR.