Some algebraic properties of differential operators

Abstract

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield K((∂⁻¹)) of pseudodifferential operators over K by the subalgebra K[∂] of all differential operators. Second, we show that the Dieudonnè determinant of a matrix pseudodifferential operator with coefficients in a differential subring A of K lies in the integral closure of A in K, and then we give an example of a 2 × 2 matrix with entries in A[∂] whose Dieudonnè determinant does not lie in A.