Nontrivial semidirect product of Lie rings of prime order

Definition
Suppose $$p$$ is a prime number. The nontrivial semidirect product of Lie rings of prime order $$p$$ is a Lie ring of order $$p$$, defined as the external semidirect product $$N \rtimes H$$, where:


 * $$N$$ is an abelian Lie ring whose additive group is the cyclic group of prime order.
 * $$H$$ is an abelian Lie ring whose additive group is the cyclic group of prime order (so that $$N$$ and $$H$$ are isomorphic). Fix an additive generator $$h$$ of $$H$$.
 * The action of $$H$$ on $$N$$ is given by multiplication maps, as follows: the element $$mh$$ of $$H$$ acts on $$N$$ by the map $$x \mapsto mx$$. Note that this map is well defined because $$m$$ is well defined modulo $$p$$.