On Rings of Weak Global Dimension at Most One

Askar Tuganbaev

doi:10.3390/math9212643

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 xy=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.

Highlights

We consider only nonzero associative unital rings
R ≤ 1 if R is a ring of weak global dimension at most one, i.e., R satisfies the following equivalent (The equivalence of the conditions is well known; e.g., see the conditions in [1] (Theorem 6.12))
Since every projective module is flat, any right or lefthereditary ring is of weak global dimension at most one. (a module M is said to be hereditary if all submodules of M are projective.) We recall that a ring R is of weak global dimension zero if and only if R is a Von Neumann regular ring, i.e., r ∈ rRr for every element r of R

Summary

Introduction

We consider only nonzero associative unital rings. For a ring R, we write w.gl.dim. R ≤ 1 if R is a ring of weak global dimension at most one, i.e., R satisfies the following equivalent (The equivalence of the conditions is well known; e.g., see the conditions in [1] (Theorem 6.12)). Since every projective module is flat, any right or left (semi)hereditary ring is of weak global dimension at most one. (a module M is said to be hereditary (resp., semihereditary) if all submodules (resp., finitely generated submodules) of M are projective.) We recall that a ring R is of weak global dimension zero if and only if R is a Von Neumann regular ring, i.e., r ∈ rRr for every element r of R.

Results

Conclusion