Lviv Algebraic Seminar – supervised by prof. V. M. Petrychkovych and prof. A. I. Gatalevych

May 17, 2022

Reduction of matrices over Bezout rings and related problems of the theory of rings and modules

Andriy Gatalevych

Associate Professor
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

The report is devoted to the study of diagonal reduction of matrices over different classes of Bezout rings of finite stable range. In terms of K-theory, the conditions are indicated when an arbitrary commutative Bezout domain is an elementary divisor ring. Semihereditary Bezout rings of Gelfand range 1 are investigated. The known theorems for Bezout rings of finite Krull dimension are generalized. The results are also obtained for the case of noncommutative Bezout rings, which are related to the structural properties of the rings. It has been proved that the commutative Bezout domain in which an arbitrary nonzero prime ideal is contained in the unique maximal ideal, is an elementary divisor ring. The notion of stable range indicates the conditions when an arbitrary commutative Bezout domain is an elementary divisor ring.

November 30, 2021

On the second and weakly-second spectrum of a module

Marta Maloid-Glebova

Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

In this talk, we study second modules over associative rings and give some basic properties of this concept. Also we define the notion of weakly-second submodule of a module over an arbitrary ring and study some relationships between second spectrum and weakly-second spectrum of a module.

October 19, 2021

A generating solution of a linear equation and structure of elements of the Zelisko group II

Volodymyr Shchedryk

Department of Algebra,
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

Victor
Bovdi

United Arab Emirates University,
Al Ain, UAE

In our reports, we will describe the structure of elements of the Zelisko group (the group of matrices which quasi-commuting with a given diagonal matrix) over a homomorphic image of a commutative Bezout domain of stable range 1.5.

May 25, 2021

About the second spectrum of a multiplication module

Marta Maloid-Glebova

Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

Let $R$ be a associative ring and $M$ an multiplicative $R$ -module. Let $Spec^s(M)$ be the the collection of all second submodules of $M$ . In this talk, we consider a new topology on $Spec^s(M)$ , called the second classical Zariski topology, and investigate the interplay between the module theoretic properties of $M$ and the topological properties of $Spec^s(M)$ . Moreover, we study $Spec^s(M)$ from point of view of spectral space.

April 28, 2021

On the classification of symmetry reductions for the (1+3)-dimensional Monge-Ampère equation

Vasyl’
Fedorchuk

Volodymyr
Fedorchuk

Department of Algebra,
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the NAS of Ukraine

The classification of symmetry reductions for the Monge – Ampère equation in the space $M(1,3) \times R(u)$ is carried out. Some results obtained by using the classification of three-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group $P(1,4)$ are presented. Here: $M(1,3)$ is the $(1+3)$ – dimensional Minkowski space; $R(u)$ is the real number axis of the dependent variable $u$ .

March 30, 2021

Comaximal factorization in a commutative Bezout ring

Oleh Romaniv

Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

We study an analogue of unique factorization rings in the case of an elementary divisor domain.

February 23, 2021

A generating solution of a linear equation and structure of elements of the Zelisko group

Volodymyr Shchedryk

Department of Algebra,
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

Victor
Bovdi

United Arab Emirates University,
Al Ain, UAE

Solutions of a linear equation $a= bx$ in a homomorphic image of a commutative Bezout domain of stable range 1.5 is developed. It is proved that the set of solutions of a solvable linear equation contains at least one solution that divides the rest, which is called a generating solution. Generating solutions are pairwise associates. Using this result, the structure of elements of the Zelisko group is investigated.

December 24, 2020

Adequate elements and domains which is not of stable range 1

Oleh Romaniv

Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

Let $R$ be a $J$ -Noetherian Bezout domain which is not a ring of stable range 1. Then in $R$ there exists a nonunit adequate element.

November 27, 2020

On adequacy of full matrices over adequate rings

Volodymyr Shchedryk

Department of Algebra,
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

Andrii
Gatalevych

Department of Higher Mathematics
Faculty of Mechanics and Mathematics Ivan Franko National University of Lviv

The set of full matrices (i.e., matrices whose elements are relatively prime) of the second order over an adequate ring $R$ is investigated. The concept of an adequate element in non-commutative rings is introduced. It is proved that nonsingular full second order matrices are right (left) adequate elements in the ring of second order matrices over $R$ .