Divergence-free polynomial derivations

In this paper we present some new and old properties of divergences and divergence-free derivations. Throughout the paper all rings are commutative with unity. Let k be a ring and let d be a k-derivation of the polynomial ring k[X] = k[x1, . . . , xn]. We denote by d the divergence of d, that is, d = ∂d(x1) ∂x1 + · · ·+ ∂d(xn) ∂xn . The derivation d is said to be divergence-free if d = 0. 1. Preliminaries Let k be a ring, and let R be a k-algebra. A k-linear mapping d : R→ R is said to be a k-derivation of R if d(ab) = ad(b) + d(a)b, for all a, b ∈ R. We denote by Derk(R) the set of all k-derivations of R. If d, d1, d2 ∈ Derk(R) and x ∈ R, then the mappings xd, d1 + d2 and [d1, d2] = d1d2 − d2d1 are also k-derivations of R. Thus, the set Derk(R) is an R-module which is also a Lie algebra. We denote by R the kernel of d, that is, R = { a ∈ R; d(a) = 0 } This set is a subring of R, called the ring of constants of R (with respect to d). If R is a field, then R is a subfield of R. 2010 Mathematics Subject Classification. Primary 12H05; Secondary 13N15.