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. ==Notes==