# Derivation equals endomorphism for Lie ring iff it is abelian

From Groupprops

## Contents

## Statement

The following are equivalent for a Lie ring :

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

## Related facts

- Inner derivation implies endomorphism for class two Lie ring
- Fully invariant implies ideal for class two Lie ring

## Proof

### (1) implies (2)

If is an abelian Lie ring, then the Lie bracket on 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 to itself. Thus, the following are equivalent for a function from to itself:

- It is a derivation of .
- It is an endomorphism of the underlying abelian group of .
- It is an endomorphism of 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 . Then, for any , we have, by the Leibniz rule:

.

This simplifies to , so is abelian.