2019
Том 71
№ 10

All Issues

Zabavskii B. V.

Articles: 14
Brief Communications (Ukrainian)

A Sharp Bézout Domain is an Elementary Divisor Ring

Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 284–288

We prove that a sharp Bézout domain is an elementary divisor ring.

Brief Communications (Ukrainian)

Rings of almost unit stable rank 1

Vasyunyk I. S., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 840-843

We introduce the notion of a ring of almost unit stable rank 1 as generalization of a ring of unit stable rank 1. We prove that the ring of almost unit stable rank 1 with the nonzero Jacobson radical is a ring of unit stable rank 1 and is also a 2-good ring. We introduce the notion of an almost 2-good ring. We show that an adequate domain is an almost 2-good ring.

Brief Communications (Ukrainian)

2-Simple ore domains of stable rank 1

Domsha O.V., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1436–1440

It is known that a simple Bézout domain is a domain of elementary divisors if and only if it is 2-simple. We prove that, over a 2-simple Ore domain of stable rank 1, an arbitrary matrix that is not a divisor of zero is equivalent to a canonical diagonal matrix.

Brief Communications (Ukrainian)

Singularities of the structure of two-sided ideals of a domain of elementary divisors

Bilyavs’ka S. I., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 854 – 856

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.

Brief Communications (Ukrainian)

Block-diagonal reduction of matrices over an $n$-simple Bézout domain $(n ≥ 3)$

Domsha O.V., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 275–280

It is known that a simple Bézout domain is the domain of elementary divisors if and only if it is 2-simple. The block-diagonal reduction of matrices over an $n$ -simple Bézout domain $(n ≥ 3)$ is realized.

Brief Communications (Ukrainian)

On the stable range of matrix rings

Petrichkovich V. M., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1575-1578

It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated.

Article (Ukrainian)

Reduction of Matrices over Bezout Rings of Stable Rank not Higher than 2

Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 550-554

We prove that a commutative Bezout ring is an Hermitian ring if and only if it is a Bezout ring of stable rank 2. It is shown that a noncommutative Bezout ring of stable rank 1 is an Hermitian ring. This implies that a noncommutative semilocal Bezout ring is an Hermitian ring. We prove that the Bezout domain of stable rank 1 with two-element group of units is a ring of elementary divisors if and only if it is a duo-domain.

Brief Communications (Ukrainian)

Factorial Analog of Distributive Bezout Domains

Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1564-1567

We investigate Bezout domains in which an arbitrary maximally-nonprincipal right ideal is two-sided. In the case of At(R) Bezout domains, we show that an arbitrary maximally-nonprincipal two-sided right ideal is also a maximally-nonprincipal left ideal.

Article (Ukrainian)

Rings with Elementary Reduction of Matrices

Romaniv A. M., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1641-1649

We establish necessary and sufficient conditions under which a quasi-Euclidean ring coincides with a ring with elementary reduction of matrices. We prove that a semilocal Bézout ring is a ring with elementary reduction of matrices and show that a 2-stage Euclidean domain is also a ring with elementary reduction of matrices. We formulate and prove a criterion for the existence of solutions of a matrix equation of a special type and write these solutions in an explicit form.

Brief Communications (Russian)

A noncommutattve analog of the cohen theorem

Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 707-710

By using weakly primary right ideals, we prove an analog of the Cohen theorem for rings of principal right ideals.

Brief Communications (Ukrainian)

Generalized adequate rings

Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 554-557

We introduce a new class of rings of elementary divisors which generalize adequate rings. We show that the problem of whether every commutative Bezout domain is a domain of elementary divisors reduces to the case where the domain contains only trivial adequate elements (namely, the identities of the domain).

Article (Ukrainian)

On noncommutative rings with elementary divisors

Zabavskii B. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 847–850

Article (Ukrainian)

Noncommutative elementary divisor rings

Zabavskii B. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 440–444

Article (Ukrainian)

Reduction of a pair of matrices over an adequate ring to a special triangular form by means of the same one-sided transformations

Kazimirskii P. S., Zabavskii B. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 256 - 258