On the differential simplicity of affine rings

Abstract

We prove that every complex regular affine ring is differentially simple relative to a set with only two derivations. The study of the differential simplicity of commutative rings has known a resurgence of interest in recent years, but its basic results go back at least to the 1950s. In order to review the results that seem most relevant to the theme of this note we introduce a few definitions. Let K be a commutative ring. A derivation d of a (commutative) K-algebra R is an endomorphism of the additive group of R that satisfies d(K) = 0 and Leibniz’s rule for the differentiation of a product, namely d(ab) = ad(b) + bd(a) for all a, b ∈ R. Denoting by DerK(R) the set of all derivations of R, let ∅ 6= D ⊂ DerK(R) be a family (finite or not) of derivations of R. An ideal I of R is D-stable if d(I) ⊂ I for all d ∈ D. Of course the ideals 0 and R are always D-stable. If R has no other D-stable ideal it is called D-simple (or differentially simple if D = DerK(R)). The 1960s saw a flurry of results on differentially simple rings, among them the classification of differentially simple algebras that are finite dimensional over a field [2] or that are affine over an algebraically closed field of positive characteristic [9]. From our point of view the most interesting result of that decade was Seidenberg’s proof in [14, Theorem 3, p. 26] that every regular affine ring over a field of characteristic zero is differentially simple. In the 1970s some of the most impressive results in the area concerned rings that are differentially simple relative to a one element family {d}. To simplify the notation these rings are called d-simple and the corresponding derivation d is said to be simple. In [10] R. Hart proved that the local ring at a nonsingular point of an irreducible variety over a field of characteristic zero is always d-simple. In the same paper he exhibited an example of a regular affine Q-algebra R that is not d-simple, for any choice of d ∈ DerK(R). On the other hand, George Bergman (unpublished) showed that the ring of polynomials Q[x, y] is d-simple for an appropriately chosen derivation d. More examples of simple derivations have appeared since then in [1], [13], [5], [3] and [6]. For some recent applications in noncommutative algebra see [4], [11] and [8]. Date: November 28, 2011. 1991 Mathematics Subject Classification. Primary: 37F75, 16S32 ; Secondary: 37J30, 32C38, 32S65.