J. F. Rittt introduced the idea of irreducible system of algebraic differential equations and showed that every system of such equations is equivalent to a finite set of irreducible systems. One of the objects of this paper is to develop a special type of abstract ideal theory which has Ritt's theorem as a consequence. The elements of our ideals are polynomials in unknowns yi, *. * , y,, and a certain number of their derivatives. Following Ritt, we call these polynomialsforms. The coefficients in these forms are assumed to be elements of a differential field 5 of characteristic zero.: A differentialfield is a commutative field (as in abstract algebra) whose elements a, b, ... have unique derivatives a,, bi, . . . which are elements of the field. These derivatives must satisfy the rules (a+b)i=al+bi and (ab)1 = alb +abi. ? The totality of these forms with coefficients in f5 is a differential ring 1. II We consider differential ideals, which are ideals containing together with any element its derivative.? An example given by Ritt shows that there exists a differential ideal of E. having no finite subset, such that every element of the ideal is a linear combination of elements of the subset and their derivatives with forms of 1R. as coefficients.** Certain results of Ritt suggested that we consider, as our purpose permits, only differential ideals which have the property that if they contain an ele-