Idealizer - Alchetron, The Free Social Encyclopedia

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 ∈ S ∣ s T ⊆ T and T s ⊆ T }

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 ∈ L ∣ [ r , S ] ⊆ 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 mention 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 S 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 ∈ R ∣ r T ⊆ T }

if T is a right ideal, or

I R ( L ) = { r ∈ R ∣ L r ⊆ 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 an R module M, the conductor or transporter is given by

( A : B ) := { r ∈ R ∣ B r ⊆ A } .

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.

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

Idealizer Wikipedia

(Text) CC BY-SA