# Shchedrik V. P.

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

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 $\Phi = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g} (\varphi 1, \varphi 2,..., \varphi n), \varphi 1| \varphi 2| ... | \varphi n, \varphi n \not = 0$.

### Bezout Rings of Stable Range 1.5

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.

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

Romaniv A. M., Shchedrik V. P.

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.

### Commutative domains of elementary divisors and some properties of their elements

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.

### A criterion for isolating a real factor from a matrix polynomial

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