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

Let FF be a field and AA a free associative FF-algebra or a group algebra of a free group with an infinite set XX of generators. We find a necessary and sufficient condition for the inclusion I′⊆II′⊆I, where I=I1…IkI=I1…Ik and I′=I1′…Il′ are any products of TT-ideals in AA. A canonical reformulation in terms of products of varieties of representations of groups answers a question posed in 1986 [A.Yu. Olshanskii, An addition to the Neumanns–Shmel’kin theorem, Matem., Mekhan. Vestnik Moskovskogo Universiteta 6 (1986) 61–64 (in Russian)].