Result: Derivations and polynomial rings

The authors consider some problems related to derivations of commutative rings. Their main results are: (1) Suppose the commutative ring \(R\) contains the rationals and that \(d_ 1,...,d_ n\) are commuting locally nilpotent derivations of \(R\). Suppose there are elements \(x_ 1,...,x_ n\) with \(A=[d_ i(x_ j)]\) invertible, and that every \(d_ i\) vanishes on every entry of \(A^{-1}\). Then \(R\) is a polynomial ring in \(n\) variables. (2) Suppose that \(R=K[x_ 1...x_ n]\) is a polynomial ring in \(n\) variables over a characteristic zero field \(K\). Let d be a derivation of \(R\) with \(d(K)=0\) and each \(d(x_ i)\) in \(K\). Then the ring of constants in \(R\) is a polynomial ring (over \(K\)) in \(n-1\) variables.