## On the length of the algebra of upper-triangular matrices.(English. Russian original)Zbl 1140.16306

Russ. Math. Surv. 60, No. 5, 984-985 (2005); translation from Usp. Mat. Nauk 60, No. 5, 177-178 (2005).
From the introduction: The problem of calculating the length of the full matrix algebra as a function of the order of matrices was first posed by A. Paz [in Linear Multilinear Algebra 15, 161-170 (1984; Zbl 0536.15007)] and still remains open. In the present paper we prove that the length of the algebra of upper-triangular matrices of order $$n$$ and of certain subalgebras of it is equal to $$n-1$$, obtain exact upper and lower estimates of the length of direct sums of finite-dimensional algebras, and obtain upper and lower estimates of the length of block-triangular matrix subalgebras.

### MSC:

 16S50 Endomorphism rings; matrix rings 16P10 Finite rings and finite-dimensional associative algebras 15A30 Algebraic systems of matrices 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)

Zbl 0536.15007
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