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It is shown that by eliminating duality theory of vector spaces from a recentproof of Kouba (O. Kouba, A duality based proof of the CombinatorialNullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a directproof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz forpolynomials over an arbitrary integral domain. The proof relies on Cramer'srule and Vandermonde's determinant to explicitly describe a map used by Koubain terms of cofactors of a certain matrix. That the CombinatorialNullstellensatz is true over integral domains is a well-known fact which isalready contained in Alon's work and emphasized in recent articles of Michalekand Schauz; the sole purpose of the present note is to point out that not onlyis it not necessary to invoke duality of vector spaces, but by not doing so oneeasily obtains a more general result.

Proof of the Combinatorial Nullstellensatz over Integral Domains, in the Spirit of Kouba