On Rings of Weak Global Dimension at Most One

Askar Tuganbaev
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Askar Tuganbaev: Power Engineering Institute, National Research University, 125252 Moscow, Russia

Mathematics, 2021, vol. 9, issue 21, 1-3

Abstract: A ring R is of weak global dimension at most one if all submodules of flat R -modules are flat. A ring R 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 R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element x ? R , there exist two nonzero central elements y , z ? R with x y = z . In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.

Keywords: ring of weak global dimension at most one; centrally essential ring; arithmetical ring; right distributive ring; left distributive ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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