Volume 221, Nº 3 (2017)

It is proven that if R is a commutative Bézout ring of Krull dimension 1, with stable range 2, then R is an elementary divisor ring.

The paper represents a series of comments to the K. A. Zhevlakov and I. P. Shestakov theorem on the existence of a locally finite in the sense of Shirshov over an ideal of the ground ring radical on the class of algebras that are algebraic over this ideal and belong to some sufficiently good homogeneous variety. It is shown in detail how the given theorem includes Plotkin’s and Kuz’min’s theorems on the existence of a locally finite radical on the classes of algebraic Lie and Mal’tsev algebras. There is adduced its generalization to locally finite extensions of ideally algebraic Lie and alternative algebras.

We consider the question of the determination of subgroups A and B such that A∩B^g ≠ 1 for any g ∈ G for a finite almost simple group G and its primary subgroups A and B of odd order. We prove that there exist only four possibilities for the ordered pair (A,B).

The construction and study of the orthogonal completion functor is an important step in the orthogonal completeness theory developed by K. I. Beidar and A. V. Mikhalev. The research of the graded orthogonal completion begun by the author is continued in this work. We consider associative rings graded by a group and modules over such rings graded by a polygon over the same group. Note that the graduation of a module by a group is a partial case of a more general and natural construction.

For any topology ℱ of a graded ring R consisting of graded right dense ideals and containing all two-sided graded dense ideals, the functor O_ℱ^gr of the graded orthogonal completion is constructed and studied in this paper. This functor maps the category of right graded R-modules into the category of right graded O_ℱ^gr(R)-modules. The important feature of the graded case is that the graded modules O_ℱ^gr(M) and O_ℱ^gr (M) (where M is a right graded R-module) may not be orthogonal complete. A criterion for the orthogonal completeness is proved. As a corollary we get that these modules are orthogonal complete in the case of a finite polygon. The properties of the functor O_ℱ^gr and a criterion of its exactness are also established.

It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N₂ of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N₂) of subvarieties of Sl w N₂ is still unknown. In our paper, we show that the lattice L(Sl w N₂) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.

In this paper, we propose a universal algorithm designed for solving large sparse linear systems over finite fields with a large prime number of elements. Such systems arise in the solution of the discrete logarithm problem modulo a prime number. The algorithm has been developed for parallel computing systems with various parallel architectures and properties. The new method inherits the structural properties of the Lanczos method. However, it provides flexible control over the complexity of parallel computations and the intensity of exchanges.