= Differential operator relative to a base ring
{title2=$D_k^m(R)$}
For a commutative $k$-algebra $R$, a relative differential operator is a $k$-linear endomorphism. Set $D_k^{-1}(R)=0$, and define order at most $m$ by $[P,m_a]\in D_k^{m-1}(R)$ for every $a\in R$, where $m_a$ means multiplication by $a$. Order zero is multiplication by an element of $R$. The finite-order operators form the ring $D_k(R)$ under composition. Specifying the base is essential: absolute operators on a complex polynomial ring include derivations of the coefficient field and need not be complex-linear.
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