{"title":"Some Certain Decompositions of Matrices Over Local Rings","authors":"M. H. Bien, P. T. Nhan, N. H. T. Nhat","doi":"10.1142/s0219498825500884","DOIUrl":null,"url":null,"abstract":"Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0219498825500884","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.
期刊介绍：
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.
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