ON THE SEMIGROUPS OF FULLY INVARIANT IDEALS OF THE FREE GROUP-ALGEBRAAND 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 and RF = (I and RF) x (J and RF) need not be true. A similar result is valid for full y invariant ideals of the free associative algebra. This implies that the product of pure varieties of group representations over an integra l domain need not be pure, that there exist pure nonprojective varieti es of group representations and of associative algebras, and also answ ers some other questions raised in the literature.