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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.

For the pairs of quasi-commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators.

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.

Let M n ( F ) be the algebra of n × n matrices over the field F and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M n ( F ) cannot exceed 2 n − 2 . We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2 n − 2 is achieved.

We describe matrices with extremal generalized centralizers over algebraically closed fields.

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.

Let M n ( F ) be the algebra of n × n matrices and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

Linear algebraic version of celebrated Double Centralizing Theorem states that the set of matrices commuting with all matrices from a centralizer of a given matrix [Formula: see text] coincides with the set of polynomials in [Formula: see text]. We examine the existence of an analogue of this classical result once commutativity is substituted by commutativity up to a factor, which is an important relation in quantum physics.