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.
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 an 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.
- 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
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