Retracts that are kernels of locally nilpotent derivations

Let k be a field of characteristic zero and B a k -domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B . We show that if B = R ⊕ I for some principal ideal I (in particular, if B is a UFD), then B = R [1] , i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k -UFD B is the kernel of two commuting locally nilpotent derivations of B , then does it follow that B ≅ R [2] ? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski’s cancellation problem and the Jacobian conjecture in affine algebraic geometry.