Differential rings used in Shirokov algebras

A classification is given of differential algebras that include $r^\alpha$, $\delta(x)$, $x\in R^n$, $\alpha=0,±1,±2,\dots$ and satisfy the condition $\varphi\psi=0$ for elements $\varphi$, $\psi$: $\operatorname{supp}\varphi\subseteq\{0\}$, $\operatorname{supp}\psi\subseteq\{0\}$.