= Algebraic differential operator ring
{title2=$D_k(R)=\bigcup_{j\geq0}D_k^j(R)$}
For a <commutative algebra> $R$ over $k$, put $D_k^{-1}(R)=0$ and define $D_k^j(R)$ to be the $k$-linear <endomorphisms> $\theta$ with $[\theta,m_a]\in D_k^{j-1}(R)$ for every multiplication map $m_a$. The union is a ring under composition, with $D_k^0(R)=R$ and $D_k^rD_k^s\subseteq D_k^{r+s}$. This commutator definition applies to singular rings and <fields> of arbitrary characteristic, unlike a definition restricted to polynomial coefficients and partial derivatives.
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