Sm. Vovsi, ON THE SEMIGROUPS OF FULLY INVARIANT IDEALS OF THE FREE GROUP-ALGEBRAAND THE FREE ASSOCIATIVE ALGEBRA, Proceedings of the American Mathematical Society, 119(4), 1993, pp. 1029-1037
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.