Derivation-invariance is centralizer-closed

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This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant subring) satisfying a Lie subring metaproperty (i.e., centralizer-closed Lie subring property)
View all Lie subring metaproperty satisfactions | View all Lie subring metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for Lie subring properties
Get more facts about derivation-invariant subring |Get facts that use property satisfaction of derivation-invariant subring | Get facts that use property satisfaction of derivation-invariant subring|Get more facts about centralizer-closed Lie subring property
ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity is centralizer-closed.
Another analogue to the same fact, in the same new context, is: characteristicity is centralizer-closed for Lie rings
View other analogues of characteristicity is centralizer-closed|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

Verbal statement

The centralizer of a derivation-invariant subring of a Lie ring is also derivation-invariant.

Related facts

Weaker facts

Proof

Given: A Lie ring L, a derivation-invariant subring S of L. C = C_L(S) is the set of a \in L such that [a,s] = 0 for all s \in S.

To prove: C is also derivation-invariant, i.e., d(C) \subseteq C for any d \in \operatorname{Der}(L).

Proof: Suppose d \in \operatorname{Der}(L). For any c \in C and s \in S, we need to show that [dc,s] = 0.

For this, note that, by the Leibniz rule property of derivations:

d([c,s]) = [dc,s] + [c,ds].

Since [c,s] = 0, the left side is zero. Further, since S is derivation-invariant, ds \in S, so [c,ds] = 0. This gives [dc,s] = 0 as required.