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A mapping of modules over a commutative ring which is an analogue of the concept of a differential operator. Let  $  R $
be a commutative ring, let  $  S $
be a subring of  $  R $
and let  $  N $
and  $  M $
be two  $  R $-
modules. A homomorphism of  $  S $-
modules  $  D :  N \rightarrow M $
is said to be a differential operator of order  $  \leq  m $, 
where  $  m $
is a non-negative integer, if for any  $  x \in R $
the mapping  $  D _ {x} :  N \rightarrow M $
defined by the formula

$$ 
D _ {x} ( n)  =  D ( xn) - x D ( n) ,
$$

is a differential operator of order  $  \leq  m - 1 $. 
A differential operator of order zero is a homomorphism of  $  R $-
modules  $  N \rightarrow M $. 
The set of all differential operators of order  $  \leq  m $
forms a submodule  $   \mathop{\rm Diff} _ {S}  ^ {m} ( N , M ) $
of the  $  R $-
module of all homomorphisms of  $  S $-
modules  $   \mathop{\rm Hom} _ {S} ( N , M ) $. 
In particular,

$$ 
 \mathop{\rm Diff} _ {S}  ^ {0} ( N , M )  \simeq   \mathop{\rm Hom} _ {R} ( N , M ) ,
$$

and the quotient module

$$ 
 \mathop{\rm Diff} _ {S}  ^ {1} ( R , M ) /  \mathop{\rm Diff} _ {S}  ^ {0} ( R , M )
$$

is isomorphic to the module of  $  S $-
derivations  $   \mathop{\rm Der} _ {S} ( R , M ) $
of  $  R $
with values in  $  M $. 
The union  $   \mathop{\rm Diff} _ {S} ( M) $
of the increasing family of submodules

$$ 
 \mathop{\rm Diff} _ {S}  ^ {0} ( M , M )  \subset    \mathop{\rm Diff} _ {S}  ^ {1} ( M ,\ 
M )  \subset  \dots
$$

is a filtered associative ring with respect to the operation of composition of mappings. This ring is known as the ring of differential operators of the  $  R $-
module  $  M $
over the subring  $  S $, 
while the corresponding graded ring

$$ 
 \mathop{\rm Symb} _ {S} ( M)  =  \oplus _ {i \geq  0 }  \mathop{\rm Symb} _ {S}  ^ {i}
( M) ,
$$

where

$$ 
 \mathop{\rm Symb} _ {S}  ^ {i} ( M)  =   \mathop{\rm Diff} _ {S}  ^ {i} ( M , M ) /
 \mathop{\rm Diff} _ {S}  ^ {i-} 1 ( M , M ) ,
$$

is said to be the module of symbols. The image of a differential operator  $  D \in  \mathop{\rm Diff} _ {S}  ^ {i} ( M , M ) $
in the ring  $   \mathop{\rm Symb} _ {S}  ^ {i} ( M) $
is said to be the symbol of the differential operator.

If  $  R $
is an algebra over the field of rational numbers and the module of differentials  $  \Omega _ {R/S}  ^ {1} $
is projective, then there exists an isomorphism between the  $  S $-
algebra  $   \mathop{\rm Diff} _ {S} ( R) $
and the enveloping algebra of the Lie algebra of  $  S $-
derivations  $   \mathop{\rm Der} _ {S} ( R , R) $. 
In this case the ring  $   \mathop{\rm Symb} _ {S} ( R) $
is isomorphic to the symmetric algebra of the  $  R $-
module  $   \mathop{\rm Der} _ {S} ( R , R ) $.

For example, let  $  R = k [ T] $
be the ring of polynomials over a field  $  k $; 
the mappings  $  \partial  / \partial  T  ^ {i} :  R \rightarrow R $, 
defined by the formula

$$ 

\frac \partial {\partial  T  ^ {i} }
 ( T  ^ {r} )  =  \left ( \begin{array}{c}
r \\
 i 
\end{array}

\right ) T  ^ {r-} i ,
$$

are differential operators of  $  R $
over  $  k $
of order  $  i $. 
The ring of differential operators  $   \mathop{\rm Diff} _ {k} ( R) $
is a free module over  $  R $
with basis  $  \partial  / \partial  T  ^ {0} \dots \partial  / \partial  T  ^ {i} ,\dots $. 
Multiplication is given by the formula

$$ 

\frac \partial {\partial  T  ^ {i} }
 \circ 
\frac \partial {\partial  T  ^ {j} }
  = \ 
\left ( \begin{array}{c}
i+ j \\
 i 
\end{array}
 \right ) 
\frac \partial {\partial  T  ^ {i+} j }
 .
$$

In particular,

$$ 
\left ( 
\frac \partial {\partial  T  ^ {1} }
 \right )  ^ {n}  =  n ! 
\frac \partial {
\partial  T  ^ {n} }

$$

(Taylor's formula) which, if the characteristic of  $  k $
is equal to zero, yields

$$ 
 \mathop{\rm Diff} _ {k} ( R)  \cong  R \left [ 
\frac \partial {\partial  T  ^ {1} }
 \right
] .
$$

If  $   \mathop{\rm Spec} ( R) $
is an affine group  $  S $-
scheme, invariant differential operators of  $  R $
may also be considered [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Vinogradov,   I.S. Krasil'shchikov,   "What is the Hamilton formalism?"  ''Russian Math. Surveys'' , '''30''' :  1  (1975)  pp. 177–202  ''Uspekhi Mat. Nauk.'' , '''30''' :  1  (1975)  pp. 173–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,   "Eléments de géométrie algébrique. IV Etude locale des schémas et des morphisms des schémas I"  ''Publ. Math. IHES'' , '''20'''  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,   P. Gabriel,   "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.E. Björk,   "The global homological dimension of some algebras of differential operators"  ''Invent. Math.'' , '''17''' :  1  (1972)  pp. 67–78</TD></TR></table>