Ideal theory and algebraic difference equations

Abstract

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-