Editing Differential operator (section)

===Relation to commutative algebra===
{{Main Article|Differential calculus over commutative algebras}}

An equivalent, but purely algebraic description of linear differential operators is as follows: an '''R'''-linear map ''P'' is a ''k''th-order linear differential operator, if for any ''k''&nbsp;+&thinsp;1 smooth functions <math>f_0,\ldots,f_k \in C^\infty(M)</math> we have

:<math>[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.</math>

Here the bracket <math>[f,P]:\Gamma(E)\to \Gamma(F)</math> is defined as the commutator

:<math>[f,P](s)=P(f\cdot s)-f\cdot P(s).</math>

This characterization of linear differential operators shows that they are particular mappings between [[module (mathematics)|modules]] over a [[commutative algebra (structure)|commutative algebra]], allowing the concept to be seen as a part of [[commutative algebra]].