FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 6, PAGES 65-173

**The length function and matrix algebras**

O. V. Markova

Abstract

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By the length of a finite system of generators for
a finite-dimensional associative algebra over an arbitrary field
we mean the least positive integer $k$ such that the words of
length not exceeding $k$ span this algebra (as
a vector space).
The maximum length for the systems of generators of an algebra is
referred to as the length of the algebra.
In the present paper, we study the main ring-theoretical properties of
the length function: the behavior of the length under unity
adjunction, direct sum of algebras, passing to subalgebras and
homomorphic images.
We give an upper bound for the length of the algebra as
a function of the nilpotency index of its Jacobson radical and
the length of the quotient algebra.
We also provide examples of the length computation for certain
algebras, in particular, for the following classical matrix
subalgebras: the algebra of upper triangular matrices, the algebra of
diagonal matrices, the Schur algebra, Courter's algebra, and for the
classes of local and commutative algebras.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k116/k11604h.htm

Last modified: November 21, 2012