A Difference-Differential Basis Theorem

Source abstract

Our aim in this paper is to extend to difference-differential rings the beautiful theorem of Kolchin [ 5 , Theorem 3] for the differential case. The necessity portion of Kolchin's result is not obtained. What might well be called the Ritt basis theorem states that if a commutative ring R with identity is finitely generated over a subring R 0 , then the ascending chain condition for radical ideals of R 0 implies the ascending chain condition for radical ideals of R . (This is indeed a basis theorem. If we define a basis for a radical ideal A to be a finite set B such that then every radical ideal of a ring R has a basis if and only if the ascending chain condition for radical ideals holds in R .) It is the Ritt basis theorem rather than the Hilbert basis theorem which has appropriate generalizations in differential and difference algebra, where in fact it originated.