LetAbe a commutative integral domain that is a finitely generated algebra over a fieldkof characteristic 0 and let ø be ak-algebra automorphism ofAof finite orderm. In this note we study the ringD(A;ø of differential operators introduced by A.D. Bell. We prove that ifAis a free module over the fixed sub-ringAø, with a basis containing 1, thenD(A;ø) is isomorphic to the matrix ringMm(D(Aø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitraryAthere is an elementcϵAsuch thatD(A[c-1];ø)≅Mm(D(A[c-1]ø)). As an application, we consider the structure ofD(A;ø)whenAis a polynomial or Laurent polynomial ring overkand ø is a diagonalizable linear automorphism.