Characteristic Lie subring not implies ideal

ANALOGY BREAKDOWN: This is the breakdown of the analogue in Lie rings of a fact encountered in group. The old fact is: characteristic implies normal.
View other analogue breakdowns of characteristic implies normal|View other analogue breakdowns from group to Lie ring

Statement

A characteristic subring of a Lie ring need not be an ideal of the Lie ring.

Related facts

Similar facts

• Characteristic not implies derivation-invariant
• Derivation-invariant not implies characteristic

Opposite facts

• Characteristic subring implies ideal in Lazard Lie ring

Analogues in other algebraic structures

• Characteristic implies normal (in groups)
• Characteristic not implies normal in loops

Proof

Example of the simple Witt algebra

Suppose $p$ is a prime number greater than 3. Let ${\mathbb{F}}_p$ be the prime field for the prime $p$. Denote by $L$ the simple Witt algebra for ${\mathbb{F}}_p$ corresponding to the prime $p$; explicitly, this means that:

• The additive group has basis $e_{-1},e_0,\dots ,e_{p-2}$. Explicitly, it is $\underset{i=-1}{\overset{p-2}{⨁}}F{e}_i$.
• The Lie bracket is defined as follows on the basis:

$[e_i,e_j]:=\{\begin{array}{cc}(j-i)e_{i+j},& -1\le i+j\le p-2\\ 0,& \\ \end{array}$

Consider the "sandwich" Lie subring $S$ of $L$ given by:

$S={⨁}_{2i>p}{\mathbb{F}}_p{e}_i$

• $S$ is clearly a subring of $L$.
• $S$ is characteristic in $L$, because $x\in S⟺[x,[x,y]]=0\mathrm{\forall }y\in L$ (note that the set with this description is not always a subring, but in this case it is).
• $S$ is not an ideal in $L$: for instance, $[S,e_{-1}]$ is not in $S$. In fact, $L$ is simple, so it has no proper nonzero ideal.