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Ideal property is centralizer-closed


Suppose L is a Lie ring, S is an ideal of S, and C = C_L(S) is the centralizer of S in L. Then, C is also an ideal of S.

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Analogue for group theory


The general version of this result is: invariance under any set of derivations is centralizer-closed, which reduces to this result when we take the inner derivations (see Lie ring acts as derivations by adjoint action). Another special case of this general result is:


Proof using the notion of inner derivations

Hands-on proof using Jacobi identity

Given: A Lie ring L, an ideal S of L. C = C_L(S) is the centralizer of S in L.

To prove: C is an ideal of L, i.e., for any x \in L and c \in C, [x,c] \in C.

Proof: To show [x,c] \in C it suffices to show that [[x,c],s] = 0 for all s \in S.

By the Jacobi identity:

[[x,c],s] + [[c,s],x] + [[s,x],c] = 0.

Since C centralizes S, [c,s] = 0, so the second term is zero. Further, since S is an ideal, [s,x] \in S, and since C centralizes S, [[s,x],c] = 0. Thus, both the second and third term on the left side are zero, forcing [[x,c],s] = 0.