On products of T-ideals in free algebras and free group algebras

Let F be a field and A a free associative F-algebra or a group algebra of a free group with an infinite set X of generators. We find a necessary and sufficient condition for the inclusion I’ into I, where I=I_1…I_k and I’=I’_1…I’_l are any products of T-ideals in A. A canonical reformulation in terms of products of group representation varieties answers a question posed in 1986