On stable T-ideals and central polynomials (Q1820847)

\textit{V. N. Latyshev} [Algebra Logika 20, 563-570 (1981; Zbl 0499.16012)] introduced the notion of stable T-ideal. This is a T-ideal U such that for every multilinear polynomial \(f(x_ 1,...,x_ n)=\sum ax_ ib\in U\) (a,b are monomials) and every \(i=1,...,n\) the polynomial \(\sum bx_ ia\) also belongs to U (here \(U\subset F<X>\)- the free associative algebra over a field F of characteristic 0). This is a generalization of a property discovered by Razmyslov for the T-ideal of the polynomial identities of the \(n\times n\) matrix algebra. The main result of the paper under review is: For every PI-algebra with a stable T-ideal of polynomial identities there exists a non-trivial central polynomial. The most important step of the proof is the following proposition which is also of independent interest: If \({\mathcal U}\) and \({\mathcal V}\) are varieties of algebras with stable T-ideals, then the same holds for \({\mathcal U}\otimes {\mathcal V}=var(A\otimes B|\) \(A\in {\mathcal U}\), \(B\in {\mathcal V})\). In particular, all T-prime and T-semiprime T-ideals are stable. (See [\textit{A. R. Kemer}, Izv. Akad. Nauk SSSR, Ser. Mat. 48, 1042-1059 (1984); English translation: Math. USSR, Izv. 25, 359-374 (1985; Zbl 0586.16010)] for this important class of T-ideals which plays an essential role in the structure theory of PI-algebras.)