# Derivation-invariant not implies characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., derivation-invariant Lie subring) neednotsatisfy the second Lie subring property (i.e., characteristic subring of a Lie ring)

View a complete list of Lie subring property non-implications | View a complete list of Lie subring property implications

Get more facts about derivation-invariant Lie subring|Get more facts about characteristic subring of a Lie ring

## Contents

## Definition

There can exist a Lie ring with a subring such that is a derivation-invariant Lie subring of , such that is not a characteristic subring of .

## Related facts

### Similar facts

- Perfect direct factor implies derivation-invariant
- Self-centralizing direct factor implies derivation-invariant

### Converse

## Facts used

## Proof

Suppose is a non-abelian Lie ring, are isomorphic copies of , and is the direct sum . Define . Then, is derivation-invariant but not characteristic.

### Proof that the subring is not characteristic

Consider the *coordinate exchange automorphism* that interchanges and . Under this automorphism, goes to . Since is non-abelian, it is not contained in , so the image of is not equal to it.

### Proof that the subring is derivation-invariant

Consider a derivation . There exist four abelian group endomorphisms that describe , namely:

.

In other words:

.

The derivation condition states that:

.

This gives:

.

We thus get:

and:

.

Setting gives that is a derivation. Setting gives that is a derivation. Plugging these back in, we get:

and:

.

Setting in the first equation gives that for all , implying that takes values in the center of . Similarly, setting in the second equation gives that is in the center of . In particular, this implies that:

takes values in .

Thus, . Since is derivation-invariant by fact (1), , so . Thus, is a derivation-invariant subring of .