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.

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.

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.

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.

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.

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.

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.

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.