Английская Википедия:Idealizer

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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.Шаблон:Sfn Such an idealizer is given by

<math>\mathbb{I}_S(T)=\{s\in S \mid sT\subseteq T \text{ and } Ts\subseteq T\}.</math>

In ring theory, if A is an additive subgroup of a ring R, then <math>\mathbb{I}_R(A)</math> (defined in the multiplicative semigroup of R) is the largest subring of R in which A is a two-sided ideal.Шаблон:SfnШаблон:Sfn

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

<math>\{r\in L\mid [r,S]\subseteq S\}</math>

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,

<math>\mathbb{I}_R(T)=\{r\in R \mid rT\subseteq T \}</math>

if T is a right ideal, or

<math>\mathbb{I}_R(L)=\{r\in R \mid Lr\subseteq L \}</math>

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

<math>(A:B):=\{r\in R \mid Br\subseteq A\}</math>.

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

<math>\mathbb{I}_R(B)=(B:B)</math>.

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.

References

• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation

Шаблон:Group-theory-stub