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.[1] Such an idealizer is given by

I S ( T ) = { s ∈ S ∣ s T ⊆ T  and  T s ⊆ T } . {\displaystyle \mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}.} {\displaystyle \mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}.}

In ring theory, if A is an additive subgroup of a ring R, then I R ( A ) {\displaystyle \mathbb {I} _{R}(A)} {\displaystyle \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.[2][3]

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 } {\displaystyle \{r\in L\mid [r,S]\subseteq S\}} {\displaystyle \{r\in L\mid [r,S]\subseteq 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 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 ∈ R ∣ r T ⊆ T } {\displaystyle \mathbb {I} _{R}(T)=\{r\in R\mid rT\subseteq T\}} {\displaystyle \mathbb {I} _{R}(T)=\{r\in R\mid rT\subseteq T\}}

if T is a right ideal, or

I R ( L ) = { r ∈ R ∣ L r ⊆ L } {\displaystyle \mathbb {I} _{R}(L)=\{r\in R\mid Lr\subseteq L\}} {\displaystyle \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 ∈ R ∣ B r ⊆ A } {\displaystyle (A:B):=\{r\in R\mid Br\subseteq A\}} {\displaystyle (A:B):=\{r\in R\mid Br\subseteq A\}}.

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

I R ( B ) = ( B : B ) {\displaystyle \mathbb {I} _{R}(B)=(B:B)} {\displaystyle \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

References

  • Goodearl, K. R. (1976), Ring theory: Nonsingular rings and modules, Pure and Applied Mathematics, No. 33, New York: Marcel Dekker Inc., pp. viii+206, MR 0429962
  • Levy, Lawrence S.; Robson, J. Chris (2011), Hereditary Noetherian prime rings and idealizers, Mathematical Surveys and Monographs, vol. 174, Providence, RI: American Mathematical Society, pp. iv+228, ISBN 978-0-8218-5350-4, MR 2790801
  • Mikhalev, Alexander V.; Pilz, Günter F., eds. (2002), The concise handbook of algebra, Dordrecht: Kluwer Academic Publishers, pp. xvi+618, ISBN 0-7923-7072-4, MR 1966155