Editing Centralizer and normalizer (section)

===Ring, algebra over a field, Lie ring, and Lie algebra===
If ''R'' is a ring or an [[algebra over a field]], and ''<math>S</math>'' is a subset of ''R'', then the centralizer of ''<math>S</math>'' is exactly as defined for groups, with ''R'' in the place of ''G''.

If <math>\mathfrak{L}</math> is a [[Lie algebra]] (or [[Lie ring]]) with Lie product [''x'', ''y''], then the centralizer of a subset ''<math>S</math>'' of <math>\mathfrak{L}</math> is defined to be{{sfn|Jacobson|1979|loc=p. 28}}

:<math>\mathrm{C}_{\mathfrak{L}}(S) = \{ x \in \mathfrak{L} \mid [x, s] = 0 \text{ for all } s \in S \}.</math>
The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If ''R'' is an associative ring, then ''R'' can be given the [[commutator#(ring theory)|bracket product]] {{nowrap|1=[''x'', ''y''] = ''xy'' − ''yx''}}. Of course then {{nowrap|1=''xy'' = ''yx''}} if and only if {{nowrap|1=[''x'', ''y''] = 0}}. If we denote the set ''R'' with the bracket product as L<sub>''R''</sub>, then clearly the ''ring centralizer'' of ''<math>S</math>'' in ''R'' is equal to the ''Lie ring centralizer'' of ''<math>S</math>''  in L<sub>''R''</sub>.

The normalizer of a subset ''<math>S</math>'' of a Lie algebra (or Lie ring) <math>\mathfrak{L}</math> is given by{{sfn|Jacobson|1979|loc=p. 28}}
:<math>\mathrm{N}_\mathfrak{L}(S) = \{ x \in \mathfrak{L} \mid [x, s] \in S \text{ for all } s \in S \}.</math>
While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the [[idealizer]] of the set ''<math>S</math>'' in <math>\mathfrak{L}</math>. If ''<math>S</math>'' is an additive subgroup of <math>\mathfrak{L}</math>, then <math>\mathrm{N}_{\mathfrak{L}}(S)</math> is the largest Lie subring (or Lie subalgebra, as the case may be) in which ''<math>S</math>'' is a Lie [[ideal (ring theory)|ideal]].{{sfn|Jacobson|1979|loc=p. 57}}