On some problems in PI-theory in characteristic \(p\) (Q2759661)

This is a survey on the author's work on verbally prime T-ideals over a field of characteristic \(p>0\) and on identities with trace. Let \(K\langle X\rangle\) be a free associative algebra of countable rank over a field \(K\). An ideal of \(K\langle X\rangle\) is a T-ideal if it is stable under all \(K\)-algebra endomorphisms of \(K\langle X\rangle\). A T-ideal \(\Gamma\) is called verbally prime if \(\Gamma_1\Gamma_2\subseteq\Gamma\) implies \(\Gamma_1\subseteq\Gamma\) or \(\Gamma_2\subseteq\Gamma\).NEWLINENEWLINENEWLINEIn his solution of the Specht problem in characteristic zero, the classification of verbally prime T-ideals was a decisive step. In characteristic \(p\) there exist T-ideals which are not finitely generated, and there seems to be little hope for a complete description of prime varieties.NEWLINENEWLINENEWLINEHowever, on the multilinear level, that is, for T-ideals generated by their multilinear elements, the author has made progress. For example, he established a relationship between the ideal structure of the finitary symmetric group and the multilinear components of the so-called classical T-ideals. The multilinear components of the verbally prime T-ideals containing all identities of the \(2\times 2\) matrix algebra over \(K\) were classified. Results on the kernel of the natural homomorphism from the ring of generic matrices over the integers to the ring of generic matrices over the field of \(p\) elements are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].