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. 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.
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 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
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