Diagonal matrix reduction over Hermite rings

A commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term 'Zabvasky subset' of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 66; no. 4; pp. 681 - 691
Main Authors Chen, Huanyin, Sheibani, Marjan
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.04.2018
Taylor & Francis Ltd
Subjects
adequate element
Elementary divisor ring
Gelfand range 1
Hermite ring
Matrix reduction
maximal ideal
Rings (mathematics)
Set theory
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Summary:A commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term 'Zabvasky subset' of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor ring if and only if with implies that there exist such that . If with implies that there exists a such that , then R is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2017.1318819