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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| Formato: | article |
| Lenguaje: | EN |
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MDPI AG
2021
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| Acceso en línea: | https://doaj.org/article/5c5cf4efd24047c2b07073b2729c4f6f |
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| Sumario: | 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. |
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