Arithmetic of matrices over rings

Shchedryk, V. P.

doi:10.15407/akademperiodika.430.278

Cited by 13 publications

(4 citation statements)

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“…It turns out that this regularity holds for a rings of stable range 1.5. Using Theorem 2.20, [10], p. 90 we get. A special role among the second-order matrices over an elementary divisor rings is played by full matrices, i.e., matrices whose elements are relatively prime.…”

Section: Theorem 3 For Any Ring R and Anymentioning

confidence: 90%

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On adequacy of full matrices

Gatalevych

Shchedryk

2023

Mat. Stud.

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This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy? The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $\mathfrak{A}$ and $\mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $\mathfrak{A}_2$ and $\mathfrak{E}_2$ denote a rings of $2 \times 2$ matrices over them. We prove that full nonsingular matrices from $\mathfrak{A}_2$ are adequate in $\mathfrak{A}_2$ and full singular matrices from $\mathfrak{E}_2$ are adequate in the set of full matrices in $\mathfrak{E}_2$.

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Section: Theorem 3 For Any Ring R and Anymentioning

confidence: 90%

“…This notion was introduced by the second author in [11] and studied in [2,10,12]. By Property 1.18, [10, p. 21], adequate rings have stable range 1.5.…”

mentioning

confidence: 99%

On adequacy of full matrices

Gatalevych

Shchedryk

2023

Mat. Stud.

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“…This definition is true in the case of quasipolynomial matrices G(x) and Φ(x), given that the elements of the form x l are invertible in the ring C[x, x −1 ]. Definition 6 (see [15]). Diagonal matrix…”

Section: Theorem On the Regularization Of Laurent Polynomial Matricesmentioning

confidence: 99%

Triangular form of Laurent polynomial matrices and their factorization

Kuchma¹,

Gatalevych²

2022

Math. Model. Comput.

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The issue of the semiscalar equivalence of Laurent polynomial matrices is investigated and the triangular form of such matrices and their finite sets is established with respect to this equivalence. The theorem on regularization of a Laurent polynomial matrix is proved. This theorem is used in the problem of factorization of such matrices. The factorization criterion of a Laurent polynomial matrix with a regular multiplier with a predetermined Smith normal form is obtained.

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“…Finally, certain properties of the Zelisko group G Φ are closely related to a factorizability of the general linear group over the ring R of stable range 1.5 (see [13,Theorem 3,p. 144], [10,Chapter 2.7], [6, Theorem 1.2.2., p. 12], [5], and [9]).…”

Section: Introductionmentioning

confidence: 99%