Reduction of matrices over Bezout rings and related problems of the theory of rings and modules
Speaker
Andriy Gatalevych
Associate Professor
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv
When & where:
May 17, 2022 at 1500, room 709
3-b Naukova Str., Lviv, Algebra Department
Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine
Abstract of talk
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.
March 1, 2022
On the second and weakly-second spectrum of a module
Speaker
Marta Maloid-Glebova
Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv
When & where:
November 30, 2021 at 1500,
online Zoom
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Meeting ID: 739 806 9872
Passcode: AUpYx5
Abstract of talk
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.
November 30, 2021
A generating solution of a linear equation and structure of elements of the Zelisko group II
Speakers
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
When & where:
October 19, 2021 at 1500,
online Zoom
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Meeting ID: 739 806 9872
Passcode: AUpYx5
Abstract of talk
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.
Victor Bovdi
October 19, 2021
About the second spectrum of a multiplication module
Speaker
MARTA MALOID-GLEBOVA
Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv
When & where:
May 25, 2021 at 1500,
online Zoom
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Meeting ID: 739 806 9872
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Abstract of talk
Let 
be a associative ring and 
an multiplicative -module. Let 
be the the collection of all second submodules of . In this talk, we consider a new topology on , called the second classical Zariski topology, and investigate the interplay between the module theoretic properties of and the topological properties of . Moreover, we study from point of view of spectral space.
May 25, 2021
On the classification of symmetry reductions for the (1+3)-dimensional Monge-Ampère equation
When & where
April 28, 2021 at 16:00
online Zoom
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Meeting ID: 739 806 9872
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Abstract of talk
The classification of symmetry reductions for the Monge – Ampère equation in the space 
is carried out. Some results obtained by using the classification of three-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group 
are presented.
Here: 
is the 
– dimensional Minkowski space; 
is the real number axis of the dependent variable 
.
Volodymyr Fedorchuk
April 28, 2021
Comaximal factorization in a commutative Bezout ring
Speaker
Oleh Romaniv
Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv
When & where:
March 30, 2021 at 1500,
online Zoom
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Meeting ID: 739 806 9872
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Abstract of talk
We study an analogue of unique factorization rings in the case of an elementary divisor domain.
March 30, 2021
A generating solution of a linear equation and structure of elements of the Zelisko group
Speakers
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
When & where:
February 23, 2021 at 1500,
online Zoom
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Meeting ID: 739 806 9872
Passcode: AUpYx5
Abstract of talk
Solutions of a linear equation 
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.
Victor Bovdi
February 23, 2021
Adequate elements and domains which is not of stable range 1
Speaker
Oleh Romaniv
Department of Algebra, Topology and Fundamental Mathematics
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv
When & where:
Decemberar 24, 2020 at 1500,
online Zoom
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Abstract of talk
Let be a 
-Noetherian Bezout domain which is not a ring of stable range 1. Then in there exists a nonunit adequate element.
December 24, 2020
On adequacy of full matrices over adequate rings
Speakers
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
When & where:
November 27, 2020 at 1500,
online Zoom
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Meeting ID: 739 806 9872
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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 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 .
Andriy Gatalevych
November 27, 2020