# Ideal not implies derivation-invariant

From Groupprops

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., ideal of a Lie ring) neednotsatisfy the second Lie subring property (i.e., derivation-invariant Lie subring)

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

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

ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: normal not implies characteristic.

View other analogues of normal not implies characteristic|View other analogues from group to Lie ring (OR, View as a tabulated list)

## Contents

## Statement

There can exist a Lie ring and a subring of such that is an ideal of and is not a derivation-invariant subring of .

## Related facts

### Similar facts for Lie rings

- Ideal property is not transitive for Lie rings
- Characteristic not implies derivation-invariant
- Characteristic ideal not implies derivation-invariant

### Related facts for groups and other algebraic structures

## Proof

Let be an abelian Lie ring whose additive group is the Klein four-group. Thus, where are subrings of size two. Since is abelian, both and are ideals of .

Let be the automorphism of that interchanges and . Then, is a derivation of , but and are not invariant under .