An upper bound for the length of a finite-dimensional algebra
Let \(A\) be a finite-dimensional algebra over a field \(F\), set \(d=\dim_FA\), and let \(e\) denote the largest degree of the minimal polynomials for the elements of \(A\). Given a finite set \(S\) containing \(1\in A\) which generates \(A\) as an \(F\)-algebra, the length \(\ell(S)\) of \(S\) is defined to be the smallest positive integer \(k\) for which \(FS^k=A\), and the length \(\ell(A)\) of \(A\) is the maximum of the numbers \(\ell(S)\), where \(S\) runs through all such finite generating subsets of \(A\). It has been shown by \textit{A. Paz} [Linear Multilinear Algebra 15, 161-170 (1984; Zbl 0536.15007)] that the length of the algebra of \(n\times n\) matrices over \(F\) is bounded by the least integer \(\geq(n^2+2)/3\). In the paper under review, the author proves that in general \(\ell(A)<e\sqrt{2d/(e-1)+1/4}+e/2-2\). When specialized to the case \(A=M_n(F)\), this gives a bound for \(\ell(A)\) of the order of magnitude of \(n^{3/2}\), a considerable improvement over the bound obtained by Paz. For special generating sets \(S\) of \(n\times n\) matrices, the bound for \(\ell(S)\) turns out to be even a linear function of \(n\). As an application of these results, a new proof is given for a theorem of \textit{L. Small, J. T. Stafford}, and \textit{R. B. Warfield} [Math. Proc. Camb. Philos. Soc. 97, 407-414 (1985; Zbl 0561.16005)] that an affine semiprime \(F\)-algebra \(A\) of Gelfand-Kirillov dimension one is a PI-algebra. In addition, the author's methods provide an upper bound for the PI-degree of \(A\).
scientific article; zbMATH DE number 1104114
scientific article