Reduction of matrices over Bezout rings and related problems of the theory of rings and modules
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.
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.
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.
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.
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 .
We study an analogue of unique factorization rings in the case of an elementary divisor domain.
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.
Let be a -Noetherian Bezout domain which is not a ring of stable range 1. Then in there exists a nonunit adequate element.
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 .