Relative Gorenstein Dimensions over Triangular Matrix Rings

Let A and B be rings, U a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><...

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Autores principales:	Driss Bennis, Rachid El Maaouy, Juan Ramón García Rozas, Luis Oyonarte
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	triangular matrix ring weakly Wakamatsu tilting modules relative Gorenstein dimensions Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/93474e7f450d4fe8815f85062b920e59
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spelling	oai:doaj.org-article:93474e7f450d4fe8815f85062b920e592021-11-11T18:14:38ZRelative Gorenstein Dimensions over Triangular Matrix Rings10.3390/math92126762227-7390https://doaj.org/article/93474e7f450d4fe8815f85062b920e592021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2676https://doaj.org/toc/2227-7390Let <i>A</i> and <i>B</i> be rings, <i>U</i> a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-bimodule, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>=</mo><mfenced open="(" close=")"><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>U</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mfenced></mrow></semantics></math></inline-formula> the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over <i>T</i> using the corresponding ones over <i>A</i> and <i>B</i>. We show that when <i>U</i> is relative (weakly) compatible, we are able to describe the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>C</mi></msub></semantics></math></inline-formula>-projective modules over <i>T</i>. As an application, we study when a morphism in <i>T</i>-Mod is a special <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>-precover and when the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a special precovering class. In addition, we study the relative global dimension of <i>T</i>. In some cases, we show that it can be computed from the relative global dimensions of <i>A</i> and <i>B</i>. We end the paper with a counterexample to a result that characterizes when a <i>T</i>-module has a finite projective dimension.Driss BennisRachid El MaaouyJuan Ramón García RozasLuis OyonarteMDPI AGarticletriangular matrix ringweakly Wakamatsu tilting modulesrelative Gorenstein dimensionsMathematicsQA1-939ENMathematics, Vol 9, Iss 2676, p 2676 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	triangular matrix ring weakly Wakamatsu tilting modules relative Gorenstein dimensions Mathematics QA1-939
spellingShingle	triangular matrix ring weakly Wakamatsu tilting modules relative Gorenstein dimensions Mathematics QA1-939 Driss Bennis Rachid El Maaouy Juan Ramón García Rozas Luis Oyonarte Relative Gorenstein Dimensions over Triangular Matrix Rings
description	Let <i>A</i> and <i>B</i> be rings, <i>U</i> a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-bimodule, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>=</mo><mfenced open="(" close=")"><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>U</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mfenced></mrow></semantics></math></inline-formula> the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over <i>T</i> using the corresponding ones over <i>A</i> and <i>B</i>. We show that when <i>U</i> is relative (weakly) compatible, we are able to describe the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>C</mi></msub></semantics></math></inline-formula>-projective modules over <i>T</i>. As an application, we study when a morphism in <i>T</i>-Mod is a special <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>-precover and when the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a special precovering class. In addition, we study the relative global dimension of <i>T</i>. In some cases, we show that it can be computed from the relative global dimensions of <i>A</i> and <i>B</i>. We end the paper with a counterexample to a result that characterizes when a <i>T</i>-module has a finite projective dimension.
format	article
author	Driss Bennis Rachid El Maaouy Juan Ramón García Rozas Luis Oyonarte
author_facet	Driss Bennis Rachid El Maaouy Juan Ramón García Rozas Luis Oyonarte
author_sort	Driss Bennis
title	Relative Gorenstein Dimensions over Triangular Matrix Rings
title_short	Relative Gorenstein Dimensions over Triangular Matrix Rings
title_full	Relative Gorenstein Dimensions over Triangular Matrix Rings
title_fullStr	Relative Gorenstein Dimensions over Triangular Matrix Rings
title_full_unstemmed	Relative Gorenstein Dimensions over Triangular Matrix Rings
title_sort	relative gorenstein dimensions over triangular matrix rings
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/93474e7f450d4fe8815f85062b920e59
work_keys_str_mv	AT drissbennis relativegorensteindimensionsovertriangularmatrixrings AT rachidelmaaouy relativegorensteindimensionsovertriangularmatrixrings AT juanramongarciarozas relativegorensteindimensionsovertriangularmatrixrings AT luisoyonarte relativegorensteindimensionsovertriangularmatrixrings
_version_	1718431899332902912

Relative Gorenstein Dimensions over Triangular Matrix Rings

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