Scrabble?! IDEALIZERS

Definitions of IDEALIZERS in various dictionaries:

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 S ( T ) = { s ∈ S ∣ s T ⊆ T and T s ⊆ T } {\displaystyle \mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}} In ring theory, if A is an additive subgroup of a ring R, then I R ( A ) {\displaystyle \mathbb {I} _{R}(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 } {\displaystyle \{r\in L\mid [r,S]\subseteq 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.