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Length computation of matrix subalgebras of special type. (English. Russian original) Zbl 1163.16017
J. Math. Sci., New York 155, No. 6, 908-931 (2008); translation from Fundam. Prikl. Mat. 13, No. 4, 165-197 (2007).
Let $$A$$ be a finite-dimensional algebra over a field $$F$$. For each set $$S\subseteq A$$ which generates $$A$$ as an $$F$$-algebra, define $$l(S)$$ to be the least integer $$h\geq 0$$ such that there is an $$F$$-basis of $$A$$ consisting of words of length $$\leq h$$ in $$S$$. The ‘length’ $$l(A)$$ of $$A$$ is the maximum of $$l(S)$$ over all finite generating sets $$S$$.
In general the length of a subalgebra of $$A$$ can be larger than the length of $$A$$. If $$A$$ is the full matrix algebra $$M_n(F)$$, then it is known that $$l(M_n(F))=O(n^{3/2})$$ and it is conjectured that $$l(M_n(F))=2n-2$$ [see C. J. Pappacena, J. Algebra 197, No. 2, 535-545 (1997; Zbl 0888.16008) and A. Paz, Linear Multilinear Algebra 15, 161-170 (1984; Zbl 0536.15007)].
In the present paper the author gives short proofs that the conjecture holds for $$n=2$$ and $$3$$, and computes the lengths of various classes of subalgebras of $$M_n(F)$$.

##### MSC:
 16S50 Endomorphism rings; matrix rings 15A30 Algebraic systems of matrices 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 16P10 Finite rings and finite-dimensional associative algebras
##### Citations:
Zbl 0888.16008; Zbl 0536.15007
Full Text:
##### References:
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