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An upper bound for the length of a finite-dimensional algebra. (English) Zbl 0888.16008

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 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 L. Small, J. T. Stafford, and 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 \(A\).

MSC:

16P10 Finite rings and finite-dimensional associative algebras
16P90 Growth rate, Gelfand-Kirillov dimension
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
15A30 Algebraic systems of matrices
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
Full Text: DOI

References:

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