Idealizer

 ( Abstract Algebra ) 

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

$\backslash mathbb\{I\}\_S(T)=\backslash \{s\backslash in\; S\; \backslash mid\; sT\backslash subseteq\; T\; \backslash text\{\; and\; \}\; Ts\backslash subseteq\; T\backslash \}.$

In ring theory, if A is an additive subgroup of a ring R, then $\backslash mathbb\{I\}\_R(A)$ (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

$\backslash \{r\backslash in\; L\backslash mid\; r,S\backslash subseteq\; S\backslash \}$

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.

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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,

$\backslash mathbb\{I\}\_R(T)=\backslash \{r\backslash in\; R\; \backslash mid\; rT\backslash subseteq\; T\; \backslash \}$

if T is a right ideal, or

$\backslash mathbb\{I\}\_R(L)=\backslash \{r\backslash in\; R\; \backslash mid\; Lr\backslash subseteq\; L\; \backslash \}$

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

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

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

$\backslash mathbb\{I\}\_R(B)=(B:B)$.

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.

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Notes

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