product of $\PI$-algebras is a $\PI$-algebra as well; secondly, to establish the existence of a sparse identity in the general case and of a Capelli identity in the finitely generated case (and thus to prove the theorem on nilpotency of the radical); and thirdly, to realize A. R. Kemer’s “supertrick” that reduces the study of identities in general algebras to that of super-identities in finitely generated superalgebras of zero characteristic. Close questions are considered in [@BP07; @Lot83; @02].

Problems related to enumeration of polylinear words which are not $n$-divisible are interesting of their own. (For example, there exists a bijection between not $3$-divisible words and Catalana numbers.) On one hand, this is a purely combinatorial problem, but on the other hand, it is related to the set of codimensions for the general matrix algebra. The study of polylinear words seems to be of great importance. V. N. Latyshev (see for instance [@LatyshevMulty]) has stated the problem of finite-basedness of the set of leading polylinear words for a $T$-ideal with respect to taking overwords and to isotonous substitutions. This problem implies the Specht problem for polylinear polynomials and is closely related to the problem of weak Noetherian property for the group 