\(\ell\)-weak identities and central polynomials for matrices
The paper deals with \(PI\)-theory. One of the most important questions is the finite basis problem. For zero characteristics (so-called Specht problem) this was done by \textit{A. R. Kemer} [Algebra Logic 27, No. 3, 167--184 (1988; Zbl 0678.16012); translation from Algebra Logika 27, No. 3, 274--294 (1988); Algebra Logika 26, No. 5, 597--641 (1987; Zbl 0646.16013)]. He also solved the problem for the local case (i.e., finitely many variables) over an infinite field [\textit{A. R. Kemer}, Math. USSR, Izv. 37, No. 1, 69--96 (1990; Zbl 0784.16016); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 4, 726--753 (1990)]. For the local case over a finite field and noetherian associative commutative ring see [\textit{A. Y. Belov}, Izv. Math. 74, No. 1, 1--126 (2010; Zbl 1208.16022); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 1, 3--134 (2010); \textit{A. Belov-Kanel} et al., Trans. Am. Math. Soc. 367, No. 8, 5553--5596 (2015; Zbl 1332.16015)]. For the case of positive characteristic, the first counterexamples were found in [\textit{A. Ya. Belov}, Fundam. Prikl. Mat. 5, No. 1, 47--66 (1999; Zbl 0964.16024)] and latter by \textit{A. V. Grishin} [Fundam. Prikl. Mat. 5, No. 1, 101--118 (1999; Zbl 1015.16022)] and \textit{V. V. Shchigolev} [Fundam. Prikl. Mat. 5, No. 1, 307--312 (1999; Zbl 0968.16008)]. However, a concrete basis of identities is known only in few cases, for example for \(2\times 2\) matrices over a field of characteristic zero [\textit{Ju. P. Razmyslov}, Algebra Logic 11, 108--120 (1973; Zbl 0266.20021)]. V. N. Latyshev introduced notion of \textit{weak identities} and established their relations with central polynomials. Although in many cases it is known that they have a finite basis, for concrete basis our knowledge is still poor. This idea generated series of papers including [\textit{A. Ya. Belov}, Sib. Mat. Zh. 44, No. 6, 1239--1254 (2003; Zbl 1054.16015); translation in Sib. Math. J. 44, No. 6, 969--980 (2003)]. For concrete basis of weak identities and central polynomials our knowledge is also still poor. In order to attack the subject, the authors develop the theory of \(\ell\)-weak identities in order to provide a feasible way of studying the central polynomials of matrix algebras. They describe the weak identities of minimal degree of matrix algebras in any dimension. They provide a description of \(\ell\)-weak identities for small degrees. This is an important step for the future theory allowing to get some concrete basis information. In this spirit, I recommend the paper [\textit{A. R. Kemer}, Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 6(325), 71--76 (1989; Zbl 0678.16013)]. For the entire collection see [Zbl 1461.16003].
scientific article; zbMATH DE number 7445607
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