Idealizer Explained

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

I_S(T)=\{s\inS\midsT\subseteqTandTs\subseteqT\}.

In ring theory, if A is an additive subgroup of a ring R, then

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\inL\mid[r,S]\subseteqS\}

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.

Comments

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,

I_R(T)=\{r\inR\midrT\subseteqT\}

if T is a right ideal, or

I_R(L)=\{r\inR\midLr\subseteqL\}

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\inR\midBr\subseteqA\}

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

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.

ExamplesThe 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.