# Derivation-invariant not implies characteristic

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., derivation-invariant Lie subring) need not satisfy the second Lie subring property (i.e., characteristic subring of a Lie ring)
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## Definition

There can exist a Lie ring $L$ with a subring $S$ such that $S$ is a derivation-invariant Lie subring of $L$, such that $S$ is not a characteristic subring of $L$.

## Facts used

1. Center is derivation-invariant

## Proof

Suppose $A$ is a non-abelian Lie ring, $A_1, A_2$ are isomorphic copies of $A$, and $L$ is the direct sum $A_1 \oplus A_2$. Define $S = A_1 +Z(L)$. Then, $S$ is derivation-invariant but not characteristic.

### Proof that the subring is not characteristic

Consider the coordinate exchange automorphism that interchanges $A_1$ and $A_2$. Under this automorphism, $A_1 + Z(L)$ goes to $A_2 + Z(L)$. Since $A_2$ is non-abelian, it is not contained in $Z(L)$, so the image of $A_1 + Z(L)$ is not equal to it.

### Proof that the subring is derivation-invariant

Consider a derivation $d:L \to L$. There exist four abelian group endomorphisms $d_{11}, d_{12}, d_{21}, d_{22}$ that describe $d$, namely: $d(x,0) = (d_{11}(x), d_{12}(x)), \qquad d(0,y) = (d_{21}(y),d_{22}(y))$.

In other words: $\! d(x,y) = (d_{11}(x) + d_{21}(y), d_{12}(x) + d_{22}(y))$.

The derivation condition states that: $\! d[(x,y),(x',y')] = [d(x,y),(x',y')] + [(x,y),d(x',y')]$.

This gives: $\! (d_{11}([x,x']) + d_{21}([y,y']), d_{12}([x,x']) + d_{22}([y,y'])) = ([d_{11}(x),x'] + [d_{21}(y),x']+ [x,d_{11}(x')] + [x,d_{12}(y')], [d_{12}(x),y'] + [d_{22}(y),y']) + [y,d_{21}(x')] + [y,d_{22}(y')])$.

We thus get: $\! d_{11}([x,x']) + d_{21}([y,y']) = [d_{11}(x),x'] + [d_{21}(y),x']+ [x,d_{11}(x')] + [x,d_{12}(y')]$

and: $\! d_{12}([x,x']) + d_{22}([y,y']) = [d_{12}(x),y'] + [d_{22}(y),y']) + [y,d_{21}(x')] + [y,d_{22}(y')]$.

Setting $y = y' = 0$ gives that $d_{11}$ is a derivation. Setting $x = x' = 0$ gives that $d_{22}$ is a derivation. Plugging these back in, we get: $\! d_{21}([y,y']) = [d_{21}(y),x']+ [x,d_{12}(y')]$

and: $\! d_{12}([x,x']) = [d_{12}(x),y'] + [y,d_{21}(x')]$.

Setting $y' = 0$ in the first equation gives that $[d_{21}(y),x'] = 0$ for all $x',y$, implying that $d_{21}$ takes values in the center of $A_1$. Similarly, setting $x' = 0$ in the second equation gives that $d_{12}(x)$ is in the center of $A_2$. In particular, this implies that: $\! d(x,0) = (d_{11}(x), d_{12}(x))$

takes values in $A_1 \oplus Z(A_2) = A_1 + Z(L) = S$.

Thus, $d(A_1) \subseteq S$. Since $Z(L)$ is derivation-invariant by fact (1), $d(Z(L)) \subseteq Z(L)$, so $d(S) = d(A_1) + d(Z(L)) \subseteq S + Z(L) = S$. Thus, $S$ is a derivation-invariant subring of $L$.