# Zabavskii B. V.

### A Sharp Bézout Domain is an Elementary Divisor Ring

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

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

### Rings of almost unit stable rank 1

Vasyunyk I. S., Zabavskii B. V.

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.

### 2-Simple ore domains of stable rank 1

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.

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

Bilyavs’ka S. I., Zabavskii B. V.

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.

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

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.

### On the stable range of matrix rings

Petrichkovich V. M., Zabavskii B. V.

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.

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

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.

### Factorial Analog of Distributive Bezout Domains

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.

### Rings with Elementary Reduction of Matrices

Romaniv A. M., Zabavskii B. V.

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.

### A noncommutattve analog of the cohen theorem

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.

### Generalized adequate rings

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).

### On noncommutative rings with elementary divisors

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

### Noncommutative elementary divisor rings

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

### 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.

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