Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings

The first part of the paper will describe a recent result of Retert in (2006) for and . This result states that if is a set of commute -derivations of such that both and the ring is -simple, then there is such that is -simple. As for applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations. 1. Introduction Let be a field of characteristic zero and denote either the ring of polynomials over or the ring of formal power series over . A -derivation of is a -linear map such that for any . Denoting by the set of all -derivations of , let be a nonempty family of -derivations. An ideal of is called -stable if for all . For example, the ideals 0 and are always -stable. If has no other -stable ideal it is called -simple. When , is often called a simple derivation. The commuting derivations have been studied by several authors: Li and Du [1], Maubach [2], Nowicki [3], Petravchuk [4], Retert [5], Van den Essen [6]. For example, it is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector; in [4], Petravchuk proved an analogous statement for derivation of over any field of characteristic zero. More explicitly, if two derivations of are linearly independent over and commute, then they have a common Darboux polynomial or they are Jacobian derivations; in [1], the authors proved the same result for and . However, we observe that this result has already been proved by Nowicki in (Now86) for both rings. Another interesting result was proved by Nowicki in [3, Theorem？？5] which says that the famous Jacobian conjecture in is equivalent to the assertion that every commutative basis of is locally nilpotent. Let be a set of commute -derivations of ; then is -simple if and only if it is -simple for some -derivation (see [5, Corollary？？2.10]). In , as pointed out in [5], up to scalar multiples, these are only sets of two commuting, nonsimple -derivations such that both and is -simple. Motivated by this, we analyze this result in [5] for and then we propose some connections with known results on commutative bases of derivations in . More precisely, the derivations are not simple -derivations of ; however, as will be shown, they can be part of a set of commuting, nonsimple -derivations such that is -simple. A trivial example is . Using the notations in [3], we give a nontrivial commutative base containing only nonsimple -derivations of the free -module such that is -simple and if the Jacobian conjecture is true in , as a consequence of