Differential Operators and Differential Modules

Abstract

In this chapter k is a differential field such that its subfield of constants C is different from k and has characteristic 0. The skew (i.e., noncommutative) ring D :=k[∂] consists of all expressions L :=a _n ∂ⁿ + ⋯ + a₁∂ + a₀ dot with n ∈ Z, n ≥ 0 and all a _i ∈ k. These elements L are called differential operators. The degree of L deg L above is m if a _m ≠ 0 and a _i = 0 for i > m. In the case L = 0 we define the degree to be −∞. The addition in D is obvious. The multiplication in D is completely determined by the prescribed rule δa = aδ + a′. Since there exists an element a ∈ k with a′ ≠ 0, the ring D is not commutative. One calls D the ring of linear differential operators with coefficients in k.