A Note on Elementary Derivations

2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15.

Let R be a UFD containing a field of characteristic 0, and Bm = R[Y1, . . . , Ym] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B3 such that ker D is a finitely generated R-algebra then the generators of ker D can be chosen to be linear in the Yi ’s. In this paper, we prove that this does not hold for B4. We also investigate R-elementary derivations D of Bm satisfying one or the other of the following conditions: (i) D is standard. (ii) ker D is generated over R by linear constants. (iii) D is fix-point-free. (iv) ker D is finitely generated as an R-algebra. (v) D is surjective. (vi) The rank of D is strictely less than m.