We will clarify the notion of differential operators from an  R-module M to an R-module N to help clarify the proof from last time.  Additionally, For the ring R=C[x_1,x_1x_2,x_1x_2^2, x_1x_2^3] and the ideal J=(x_1x_2,x_1x_2^2), we will compute the idealizer of J, JD(R) and D(R,J) to see that the quotient of the idealizer modulo JD(R) is not equal to the idealizer modulo D(R,J).  Look forward to some interesting pictures.