Ideal not implies derivation-invariant
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) need not satisfy 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)
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
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 .