(defined in the multiplicative semigroup of R) is the largest subring of R in which A is a two-sided ideal.
Often, when right or left ideals are the additive subgroups of R of interest, the idealizer is defined more simply by taking advantage of the fact that multiplication by ring elements is already absorbed on one side.
In commutative algebra, the idealizer is related to a more general construction.
Given a commutative ring R, and given two subsets A and B of a right R-module M, the conductor or transporter is given by In terms of this conductor notation, an additive subgroup B of R has idealizer When A and B are ideals of R, the conductor is part of the structure of the residuated lattice of ideals of R. The multiplier algebra M(A) of a C*-algebra A is isomorphic to the idealizer of π(A) where π is any faithful nondegenerate representation of A on a Hilbert space H.
This group theory-related article is a stub.