##### Work number - M 81 FILED

*Authors: Bilavska-Kydlata S.I., Domsha O.V., Vasyunyk I.S.*

Presented by Ivan Franko National University of Lviv.

The cycle of scientific works consists of 23 scientific articles, published during 9 years.

Bases of theory of diagonalization of matrices are created using of one of important invariants of K-theory – stable range of ring.

The fundamental results are received: a stable range and a generalized stable range is calculation for the different classes of rings. It is indicated connection of finite homomorphic image of adequate and adequate in a zero rings with clean rings, exchange rings, the rings of idempotent stable range 1, Gelfand rings and PM-rings. Generalization of theorems of Cohen is got for the modules and non-commutative rings. Next to researches of classic diagonal reduction of matrices of rings, the new types of reduction are described, in particular, block-diagonal and reduction of matrices in localization of rings. Also the new classes of commutative and non-commutative elementary divisors rings are described. The theoretic-structural properties of matrices above the Bezout rings are investigated.

The cycle of scientific works contains priority results which touch algebra of matrices over rings, K-theory and theory of the modules.

The new approaches the study of structure of different classes of rings of finite generalizated principal ideals and connections between them, method of calculation of stable range of different classes of rings and their generalizations is proposed.

The results of cycle is important for the next study of structure of rings and matrices over rings and also in communicating sections of algebra, K-theory, theory of differential equalizations and in the applied tasks.

*The results of researches are expounded in 23 articles (from them 2 in foreign journals) and 23 theses of conferences. Works of authors are cited in more than 5 scientific and scientifically applied journals (according to database Scopus), h-index=1.*