What is "idealizer"

Idealizer \I*de"al*i`zer\, n. An idealist.

n. A person who idealizes

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

I(T) = {s ∈ S ∣ sT ⊆ Tand Ts ⊆ T}

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

{r ∈ L ∣ [r, S] ⊆ 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.