# 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

In ring theory, if *A* is an additive subgroup of a ring *R*, then (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

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,

if *T* is a right ideal, or

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

- .

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

- .

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

- Mikhalev 2002, p.30.
- Goodearl 1976, p.121.
- 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