AbstractIt is shown that by eliminating duality theory of vector spaces from a recent proofof Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J.Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-versionof Alon’s Combinatorial Nullstellensatz for polynomials over an arbitrary integraldomain. The proof relies on Cramer’s rule and Vandermonde’s determinant toexplicitly describe a map used by Kouba in terms of cofactors of a certain matrix.That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon’s work and emphasized in recentarticles of Micha lek and Schauz; the sole purpose of the present note is to point outthat not only is it not necessary to invoke duality of vector spaces, but by not doingso one easily obtains a more general result...