In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S is the set of elements of G that commute with S "as a whole". The centralizer and normalizer of S are subgroups of G, and can provide insight into the structure of G.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 Lie algebra.The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.