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

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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) need not satisfy the second Lie subring property (i.e., ideal of a Lie ring)
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Statement

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

Facts used

  1. Subring of a Lie ring that is maximal as a subgroup

Proof

Let L be the Klein four-group with x,y two of the non-identity elements. The four elements of L are 0,x,y,x+y. We define the Lie bracket as follows:

[x,y] = x

This completely determines the Lie bracket:

[x,x+y] = [y,x] = [x+y,x] = [y,x+y] = [x+y,y] = x.

Let S be the subring of L defined as \{ 0,y \}. The sum of S with any subring of L that does not contain S is the whole ring L (since S is maximal as a subgroup of L). Also, the sum of S with any subring of L contained in S is S. Thus, in any case, the sum of S with any subring is a subring. (for more, see fact (1)).

However, S is not an ideal of L, because [y,x] = x, and x is not in S.