First-order algebraic differential operator decomposition
= First-order algebraic differential operator decomposition
{title2=$D_k^1(R)=R\oplus\operatorname{Der}_k(R)$}
If $\theta$ has order at most one, set $c=\theta(1)$ and $\delta=\theta-m_c$. Then $\delta(1)=0$ and $[\delta,m_a]=m_{\delta(a)}$, which is the <Leibniz rule>. Thus $\delta$ is a <derivation of an algebra>. Conversely any derivation satisfies this commutator identity. The sum is direct because a derivation that is multiplication must vanish at $1$.