  It is shown that by eliminating duality theory of vector spaces from a recent
proof of Kouba (O. Kouba, A duality based proof of the Combinatorial
Nullstellensatz. Electron. J. Combin. 16 (2009), #N9) one obtains a direct
proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for
polynomials over an arbitrary integral domain. The proof relies on Cramer's
rule and Vandermonde's determinant to explicitly 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 recent articles of Michalek
and Schauz; the sole purpose of the present note is to point out that not only
is it not necessary to invoke duality of vector spaces, but by not doing so one
easily obtains a more general result.
