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In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order 2k+1 over an arbitrary field of characteristic 2 is equal to 2k.

The matrix relation AB = CBA is investigated. An explicit description of the space of matrices B satisfying this relation is obtained for an arbitrary fixed matrix C and a diagonalizable matrix A. The connection between this space and the family of right annihilators of the matrices A−λC, where λ ranges over the set of eigenvalues of the matrix A, is studied. In the case where AB = CBA, AC = CA, and BC = CB, a canonical form for A,B,C, generalizing Thompson’s result for invertible A,B,C, is introduced. Also bounds for the lengths of pairs of matrices {A,B} of the form indicated are provided. Bibliography: 26 titles.

A lower and an upper bounds for the length of a direct sum of nonassociative algebras are obtained, and their sharpness is established. Note that while the lower bound for the length of a direct sum in the associative and nonassociative cases turns out to be the same, the upper bound in the nonassociative case significantly exceeds its associative counterpart.

The paper evaluates the lengths of group algebras of all groups of orders not exceeding 7 over an arbitrary field.

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N n (𝔽) over a field 𝔽 with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M n (𝔽). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.

The paper continues the investigation of the lengths of quasi-commuting matrix pairs; specifically, it considers the problem of realizability of different positive integers as values of the length function for quasi-commuting matrix pairs.

The notion of graph generated by the mutual orthogonality relation for the elements of an associative ring is introduced. The main attention is paid to the commutative rings and to the matrix ring over a field and its various subrings and subsets. In particular, the diameters of the orthogonality graphs of the full matrix algebra over an arbitrary field and its subsets consisting of diagonal, diagonalizable, triangularizable, and nilpotent matrices are computed. Bibliography: 36 titles.

A matrix is said to be semimagic if all its row and column sums are equal. The paper investigates the length function for the algebra of semimagic matrices with respect to different generating systems. Bibliography: 11 titles.

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 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, passage 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 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.

We define the length of a finite system of generators of a given algebra \( \mathcal{A} \) as the smallest number k such that words of length not greater than k generate \( \mathcal{A} \) as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length 1 up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length 1, as well as those that are maximal with respect to inclusion.

Let {ie910-01} be a field and let {ie910-02} be a finite-dimensional {ie910-03}-algebra. We define the length of a finite generating set of this algebra as the smallest number k such that words of length not greater than k generate {ie910-04} as a vector space, and the length of the algebra is the maximum of the lengths of its generating sets. In this article, we give a series of examples of length computation for matrix subalgebras. In particular, we evaluate the lengths of certain upper triangular matrix subalgebras and their direct sums, and the lengths of classical commutative matrix subalgebras. The connection between the length of an algebra and the lengths of its subalgebras is also studied.