Editing Centralizer and normalizer (section)

===Rings and algebras over a field===
Source:{{sfn|Jacobson|1979|loc=p. 28}}
* Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.
* The normalizer of ''<math>S</math>'' in a Lie ring contains the centralizer of ''<math>S</math>''.
* C<sub>''R''</sub>(C<sub>''R''</sub>(''S'')) contains ''<math>S</math>'' but is not necessarily equal. The [[double centralizer theorem]] deals with situations where equality occurs.
* If ''<math>S</math>'' is an additive subgroup of a Lie ring ''A'', then N<sub>''A''</sub>(''S'') is the largest Lie subring of ''A'' in which ''<math>S</math>'' is a Lie ideal.
* If ''<math>S</math>'' is a Lie subring of a Lie ring ''A'', then {{nowrap|''S'' ⊆ N<sub>''A''</sub>(''S'')}}.