Inabstract algebra, theidealizer of a subsemigroupT of asemigroupS is the largest subsemigroup ofS in whichT is anideal.^[1] Such an idealizer is given by

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

Inring theory, ifA is an additive subgroup of aringR, then${\displaystyle {\mathbb{I}}_R(A)}$ (defined in the multiplicative semigroup ofR) is the largest subring ofR in whichA is a two-sided ideal.^[2][3]

InLie algebra, ifL is aLie ring (orLie algebra) with Lie product [x,y], andS is an additive subgroup ofL, then the set

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

is classically called thenormalizer ofS, 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, becauseanticommutativity of the Lie product causes [s,r] = −[r,s] ∈ S. The Lie "normalizer" ofS is the largest subring ofL in whichS is a Lie ideal.

Often, when right or left ideals are the additive subgroups ofR 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,

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

ifT is a right ideal, or

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

ifL is a left ideal.

Incommutative algebra, the idealizer is related to a more general construction. Given a commutative ringR, and given two subsetsA andB of a rightR-moduleM, theconductor ortransporter is given by

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

In terms of this conductor notation, an additive subgroupB ofR has idealizer

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

WhenA andB are ideals ofR, the conductor is part of the structure of theresiduated lattice of ideals ofR.

Examples

Themultiplier algebraM(A) of aC*-algebraA isisomorphic to the idealizer ofπ(A) whereπ is any faithful nondegenerate representation ofA on aHilbert space H.

• 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

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