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

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

February 23, 2021 at 15:00 online

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Abstract of talk

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.

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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

December 24, 2020 at 12:00 online

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Abstract of talk

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.

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On adequacy of full matrices over adequate rings

Andrii Gatalevych

Department of Higher Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of L’viv

Volodymyr Shchedryk

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

November 27, 2020 at 16:00 online

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Abstract of talk

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.

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With deep regret we report that on August 1, 2020 suddenly passed away

supervisor of our seminar, Honored Professor of the Ivan Franko University of Lviv,

Doctor of Sciences in Physics and Mathematics, Head of the Department of Algebra and Logic,

a famous scientist and talented teacher

prof. Bogdan V. Zabavsky

(19.08.1957 − 01.08.2020)

We express our deep and sincere condolences to his family and friends,

employees of the Department of Algebra and Logic, students.

Our memory of prof. Bogdan V. Zabavsky will never die.

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On the semiscalar equivalence of 3-by-3 matrices with all different characteristic roots

Bogdan Shavarovskii

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

October 15, 2019 at 15:05 in Lecture Room 377

Abstract of talk

Some invariants of 3-by-3 polynomial matrices with all different characteristic roots are found. In some cases, the conditions of semiscalar equivalence of such matrices are specified and canonical forms are constructed.

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Equivalence of matrices over quadratic rings and matrix equations

Natalija Ladzoryshyn

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

September 24, 2019 at 15:05 in Lecture Room 377

Abstract of talk

The notation of the $(z,k)$ -equivalence of matrices over quadratic rings is introduced. The standard forms of matrices and their pairs with respect to this equivalence are established. The conditions of solvability of matrix linear unilateral and bilateral equations over an arbitrary quadratic ring are given and their integer solutions are described. On the basis of constructed standard forms of matrices, we proposed the effective method of solvability of these matrix equations and description of the structure of their solutions.

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Equivalence and factorization of the Kronecker products of polynomial matrices

Volodymyr Zelisko

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

April 16, 2019 at 15:05 in Lecture Room 377

Abstract of talk

We consider the problem of equivalence and semiscalar equivalence of the Kronecker products of polynomial matrices and their separation of factors.

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The structure of block matrices. I. Factorizations in the rings of the block matrices

Vasyl’ Petrychkovych

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

Nataliia Dzhaliuk

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

March 12, 2019 at 15:05 in Lecture Room 377

Abstract of talk

We investigate equivalences, factorizations, and matrix unilateral and bilateral equations in the rings of the block matrices. In the first part, we consider factorizations of the block matrices, in particular, the block triangular and the block diagonal matrices over integral domains of finitely generated principal ideals. Conditions for existence and uniqueness up to the association of the factorizations in such rings are established.

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