# Lie subring whose sum with any subring is a subring not implies ideal

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., Lie subring whose sum with any subring is a subring) neednotsatisfy the second Lie subring property (i.e., ideal of a Lie ring)

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## Statement

There can exist a Lie ring and a subring , such that the sum of and any subring of is a subring, but is not an ideal.

## Facts used

## Proof

Let be the Klein four-group with two of the non-identity elements. The four elements of are . We define the Lie bracket as follows:

This completely determines the Lie bracket:

.

Let be the subring of defined as . The sum of with any subring of that does not contain is the whole ring (since is maximal as a subgroup of ). Also, the sum of with any subring of contained in is . Thus, in any case, the sum of with any subring is a subring. (for more, see fact (1)).

However, is not an ideal of , because , and is not in .