On Rings of Weak Global Dimension at Most One

On Rings of Weak Global Dimension at Most One

A ring <i>R</i> is of weak global dimension at most one if all submodules of flat <i>R</i>-modules are flat. A ring <i>R</i> is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left...

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Autor principal: Askar Tuganbaev
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
ring of weak global dimension at most one
centrally essential ring
arithmetical ring
right distributive ring
left distributive ring
Mathematics
QA1-939
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spelling oai:doaj.org-article:5c5cf4efd24047c2b07073b2729c4f6f2021-11-11T18:13:31ZOn Rings of Weak Global Dimension at Most One10.3390/math92126432227-7390https://doaj.org/article/5c5cf4efd24047c2b07073b2729c4f6f2021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2643https://doaj.org/toc/2227-7390A ring <i>R</i> is of weak global dimension at most one if all submodules of flat <i>R</i>-modules are flat. A ring <i>R</i> is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of <i>R</i> is distributive. Jensen has proved earlier that a commutative ring <i>R</i> is a ring of weak global dimension at most one if and only if <i>R</i> is an arithmetical semiprime ring. A ring <i>R</i> is said to be centrally essential if either <i>R</i> is commutative or, for every noncentral element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>, there exist two nonzero central elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi><mo>=</mo><mi>z</mi></mrow></semantics></math></inline-formula>. In Theorem 2 of our paper, we prove that a centrally essential ring <i>R</i> is of weak global dimension at most one if and only is <i>R</i> is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.Askar TuganbaevMDPI AGarticlering of weak global dimension at most onecentrally essential ringarithmetical ringright distributive ringleft distributive ringMathematicsQA1-939ENMathematics, Vol 9, Iss 2643, p 2643 (2021)
institution DOAJ
collection DOAJ
language EN
topic ring of weak global dimension at most one
centrally essential ring
arithmetical ring
right distributive ring
left distributive ring
Mathematics
QA1-939
spellingShingle ring of weak global dimension at most one
centrally essential ring
arithmetical ring
right distributive ring
left distributive ring
Mathematics
QA1-939
Askar Tuganbaev
On Rings of Weak Global Dimension at Most One
description A ring <i>R</i> is of weak global dimension at most one if all submodules of flat <i>R</i>-modules are flat. A ring <i>R</i> is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of <i>R</i> is distributive. Jensen has proved earlier that a commutative ring <i>R</i> is a ring of weak global dimension at most one if and only if <i>R</i> is an arithmetical semiprime ring. A ring <i>R</i> is said to be centrally essential if either <i>R</i> is commutative or, for every noncentral element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>, there exist two nonzero central elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi><mo>=</mo><mi>z</mi></mrow></semantics></math></inline-formula>. In Theorem 2 of our paper, we prove that a centrally essential ring <i>R</i> is of weak global dimension at most one if and only is <i>R</i> is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.
format article
author Askar Tuganbaev
author_facet Askar Tuganbaev
author_sort Askar Tuganbaev
title On Rings of Weak Global Dimension at Most One
title_short On Rings of Weak Global Dimension at Most One
title_full On Rings of Weak Global Dimension at Most One
title_fullStr On Rings of Weak Global Dimension at Most One
title_full_unstemmed On Rings of Weak Global Dimension at Most One
title_sort on rings of weak global dimension at most one
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/5c5cf4efd24047c2b07073b2729c4f6f
work_keys_str_mv AT askartuganbaev onringsofweakglobaldimensionatmostone
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