Commutative Bezout domains of stable range 1.5

Victor A. Bovdi; Volodymyr P. Shchedryk

doi:10.1016/j.laa.2018.06.012

Commutative Bezout domains of stable range 1.5

Victor A. Bovdi, Volodymyr P. Shchedryk

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

A ring R is said to be of stable range 1.5 if for each a,b∈R and 0≠c∈R satisfying aR+bR+cR=R there exists r∈R such that (a+br)R+cR=R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R.

Original language	English
Pages (from-to)	127-134
Number of pages	8
Journal	Linear Algebra and Its Applications
Volume	568
DOIs	https://doi.org/10.1016/j.laa.2018.06.012
Publication status	Published - May 1 2019

Keywords

Adequate ring
Commutative Bezout domain
Elementary divisor ring
Stable range of a ring

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2018.06.012

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AU - Bovdi, Victor A.

AU - Shchedryk, Volodymyr P.

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AB - A ring R is said to be of stable range 1.5 if for each a,b∈R and 0≠c∈R satisfying aR+bR+cR=R there exists r∈R such that (a+br)R+cR=R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R.

KW - Adequate ring

KW - Commutative Bezout domain

KW - Elementary divisor ring

KW - Stable range of a ring

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Commutative Bezout domains of stable range 1.5

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Keywords

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