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

The thesis 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. Bezout rings and their homomorphic images with conditions on the Jacobson radical are studied. 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. The influence of the spectrum of Bezout commutative rings on the possibility of diagonal reduction of matrices is investigated. One result is the following: a commutative Bezout ring with a finite number of minimal prime ideals is an elementary divisor ring if and only if when all factor rings over prime ideals are elementary divisor ring. We prove that Hermite ring whose classical ring of quotients is a Boolean ring is an elementary divisor ring. It is shown that over the commutative Bezout ring of stable range , an arbitrary row of length is complemented by a matrix whose determinant is equal to the largest common divisor of all elements of this row. The concept and properties of a Gelfand element and a ring of Gelfand range 1 are introduced. It is proved that the semihereditary Bezout ring in which an arbitrary regular element is a Gelfand element is an elementary divisor ring. Also it is proved that the commutative Bezout ring of stable range 2 and Gelfand range 1 is an elementary divisor ring. It is proved that the commutative Bezout ring of stable range 2 and dimension Krull 1 dimension is an elementary divisor ring. It is shown that in a commutative ring of Krull dimension 1 each regular element is an element of almost stable range 1. We introduce the concept of a generalized B-ring as a ring of stable range 2. The conditions under which the generalized B-ring is a ring of stable range 1 and elementary divisor ring are established. The structure of the maximal ideals of a ring with conditions on the Jacobson radical is described. The structure of the maximal ideals of the factor ring of the commutative Bezout domain is described, provided that it is a Kasch ring. It is proved that fractional -ring is fractional-regular and if its stable range does not exceed 2, it is an elementary divisor ring It is proved that the commutative Bezout domain with the nonzero principal Jacobson radical, is a ring of stable range 1. The structure of the maximal ideals of a ring with conditions on the Jacobson radical is described. The structure of the maximal ideals of the factor ring of the commutative Bezout domain is described, provided that it is a Kasch ring. It is proved that fractional -ring is fractional-regular and if its stable range does not exceed 2, it is an elementary divisor ring It is proved that the commutative Bezout domain with the nonzero principal Jacobson radical, is a ring of stable range 1. Many classes of noncommutative Bezout rings have also been studied. The properties of the elements of Bezout rings are described, in which any non-principal right ideal is two-sided. The conditions under which the sets of maximally non-principal right and left ideals coincide are indicated. The result of Asano and Nakayama on the symmetry of divisibility for the case of elementary divisor ring is generalized. It is proved that the unit-central ring of stable range 1 is a quasi-duo ring and an elementary divisor ring if and only if it is a duo ring. We introduced the notions of rings of regular (von Neumann) range 1, rings of semihereditary range 1, rings of regular range 1. We found connections between the entered ranges for abelian and duo rings. It is proved that the right Bezout ring in which the Jacobson radical contains a completely prime ideal, is the right Hermite ring. It is shown that the Bezout distributive ring, in which the Jacobson radical contains a completely prime ideal, and there are no two sided ideals other than trivial, is not an elementary divisor ring. It is proved that a pair of matrices over an adequate duo ring can be reduced to a special triangular form by identical one-sided transformations. A new class of rings -ring has been introduced and it is a generalization of -rings. Were established properties and examples of such rings. The results of the work are mainly theoretical. The obtained results and the proposed methods can be used in further research in the theory of rings and modules, K-theory, as well as in more applied problems of linear algebra.