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

Markova, O. V.

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)

Keywords:

lengths of matrix algebras; algebras of upper-triangular matrices; lengths of direct sums of finite-dimensional algebras

Citations:

Zbl 0536.15007