On the semigroups of fully invariant ideals of the free group algebra and the free associative algebra

Let $R$ be an integral domain, $K$ its field of fractions, $F$ a free group. Let $I$ and $J$ be fully invariant (=verbal) ideals of the group algebra $KF$. We prove that over certain domains the equality $IJ \cap RF = (I \cap RF) \times (J \cap RF)$ need not be true. A similar result is valid for fully invariant ideals of the free associative algebra. This implies that the product of pure varieties of group representations over an integral domain need not be pure, that there exist pure nonprojective varieties of group representations and of associative algebras, and also answers some other questions raised in the literature.