2019
Том 71
№ 2

All Issues

Shchedrik V. P.

Articles: 5
Article (Ukrainian)

Bezout rings of stable ranк 1.5 and the decomposition of a complete linear group into its multiple subgroups

Shchedrik V. P.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 113-120

A ring $R$ is called a ring of stable rank 1.5 if, for any triple $a, b, c \in R, c \not = 0$, such that $aR + bR + cR = R$, there exists $r \in R$ such that $(a + br)R + cR = R$. It is proved that a commutative Bezout domain has a stable rank 1.5 if and only if every invertible matrix $A$ can be represented in the form $A = HLU$, where $L, U$ are elements of the groups of lower and upper unitriangular matrices (triangular matrices with 1 on the diagonal) and the matrix $H$ belongs to the group $$\bf{G} \Phi = \{ H \in \mathrm{G}\mathrm{L}n(R) | \exists H_1 \in \mathrm{G}\mathrm{L}_n(R) : H\Phi = \Phi H_1\},$$ where $\Phi = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g} (\varphi 1, \varphi 2,..., \varphi n), \varphi 1| \varphi 2| ... | \varphi n, \varphi n \not = 0$.

Article (Ukrainian)

Bezout Rings of Stable Range 1.5

Shchedrik V. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 849–860

A ring $R$ has a stable range 1.5 if, for every triple of left relatively prime nonzero elements $a, b$ and $c$ in $R$, there exists $r$ such that the elements $a+br$ and $c$ are left relatively prime. Let $R$ be a commutative Bezout domain. We prove that the matrix ring $M_2 (R)$ has the stable range 1.5 if and only if the ring $R$ has the same stable range.

Brief Communications (Ukrainian)

Greatest common divisor of matrices one of which is a disappear matrix

Romaniv A. M., Shchedrik V. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 425–430

We study the structure of the greatest common divisor of matrices one of which is a disappear matrix. In this connection, we indicate the Smith normal form and the transforming matrices of the left greatest common divisor.

Article (Ukrainian)

Commutative domains of elementary divisors and some properties of their elements

Shchedrik V. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 126-139

We study commutative domains of elementary divisors from the viewpoint of investigation of the structure of invertible matrices that reduce a given matrix to the diagonal form. Some properties of elements of these domains are indicated. We establish conditions, close to the stable-rank conditions, under which a commutative Bezout domain is a domain of ´ elementary divisors.

Article (Ukrainian)

A criterion for isolating a real factor from a matrix polynomial

Shchedrik V. P.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 3. - pp. 370-373