Idealizer of a Lie subring

Definition
Let $$L$$ be a Lie ring and $$A$$ be a subring. The idealizer of $$A$$ in $$L$$ is defined in the following equivalent ways:


 * It is the largest subring of $$L$$ containing $$A$$, in which $$A$$ is an ideal.
 * It is the subring of $$L$$ comprising those $$x$$ for which $$[A,x] \subset A$$.