Ideal not implies derivation-invariant

From Groupprops
Jump to: navigation, search
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)

Statement

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

Related facts

Similar facts for Lie rings

Converse and related facts

Related facts for groups and other algebraic structures

Proof

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

Let \sigma be the automorphism of L that interchanges A and B. Then, \sigma is a derivation of L, but A and B are not invariant under \sigma.