# Idealizer

In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal. Such an idealizer is given by

$\mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}.$ In ring theory, if A is an additive subgroup of a ring R, then $\mathbb {I} _{R}(A)$ (defined in the multiplicative semigroup of R) is the largest subring of R in which A is a two-sided ideal.

In Lie algebra, if L is a Lie ring (or Lie algebra) with Lie product [x,y], and S is an additive subgroup of L, then the set

$\{r\in L\mid [r,S]\subseteq S\}$ is classically called the normalizer of S, however it is apparent that this set is actually the Lie ring equivalent of the idealizer. It is not necessary to specify that [S,r]  S, because anticommutativity of the Lie product causes [s,r] = −[r,s]  S. The Lie "normalizer" of S is the largest subring of L in which S is a Lie 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. Explicitly,

$\mathbb {I} _{R}(T)=\{r\in R\mid rT\subseteq T\}$ if T is a right ideal, or

$\mathbb {I} _{R}(L)=\{r\in R\mid Lr\subseteq L\}$ if L is a left ideal.

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

$(A:B):=\{r\in R\mid Br\subseteq A\}$ .

In terms of this conductor notation, an additive subgroup B of R has idealizer

$\mathbb {I} _{R}(B)=(B:B)$ .

When A and B are ideals of R, the conductor is part of the structure of the residuated lattice of ideals of R.

Examples

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.