Derivation equals endomorphism for Lie ring iff it is abelian

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Statement

The following are equivalent for a Lie ring L:

  1. L is an Abelian Lie ring (?), i.e., the Lie bracket on L is identically zero.
  2. A function from L to itself is a derivation of L if and only if it is an endomorphism of L.
  3. Every endomorphism of L is a derivation of L.
  4. Every automorphism of L is a derivation of L.
  5. The identity map is a derivation of L.

Related facts

Proof

(1) implies (2)

If L is an abelian Lie ring, then the Lie bracket on L is identically zero. Thus, the Leibniz rule for derivations as well as the Lie bracket-preservation condition for endomorphisms is satisfied by all functions from L to itself. Thus, the following are equivalent for a function from L to itself:

  • It is a derivation of L.
  • It is an endomorphism of the underlying abelian group of L.
  • It is an endomorphism of L as a Lie ring.

This shows that (2) holds.

(2) implies (3) implies (4) implies (5)

This is obvious.

(5) implies (1)

Suppose the identity map is a derivation of L. Then, for any x,y \in L, we have, by the Leibniz rule:

[x,y] = [x,y] + [x,y].

This simplifies to [x,y] = 0, so L is abelian.