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...
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | article |
| Langue: | EN |
| Publié: |
MDPI AG
2021
|
| Sujets: | |
| Accès en ligne: | https://doaj.org/article/5c5cf4efd24047c2b07073b2729c4f6f |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| Résumé: | 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. |
|---|