Let \(F\) be a field; let \(G_i\) be a group (\(i=1,2\)) and let \(V_i\) be an irreducible, primitive, finite-dimensional \(FG_i\)-module. Put \(G=G_1\times G_2\) and \(V=V_1\otimes_FV_2\). The main aim of this paper is to determine sufficient conditions for \(V\) to be primitive as a \(G\)-module. It turns out to be the case if \(V_1\) and \(V_2\) are absolutely irreducible and \(V_1\) is absolutely quasi-primitive. As such it extends a result of \textit{N. S. Hekster} [Indag. Math. 47, 63-76 (1985; Zbl 0559.20009)]; he showed \(V\) is primitive whenver \(|G|<\infty\) and \(F=\mathbb{C}\). The classification of the finite simple groups is used at a specific point to prove results in the paper. Furthermore, the reader should also consult an earlier but related result of the reviewer, mentioned by these authors [see Publ. Math. 35, No. 1/2, 149-153 (1988; Zbl 0668.20013)].