Idealizer

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

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

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

$\{r\in L\mid [r,S]\subseteq S\}$

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 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,

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

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 an 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

1. Mikhalev 2002, p.30.
2. Goodearl 1976, p.121.
3. Levy & Robson 2011, p.7.

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 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