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On Rings of Weak Global Dimension at Most One

Author

Listed:
  • Askar Tuganbaev

    (Power Engineering Institute, National Research University, 125252 Moscow, Russia)

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.

Suggested Citation

  • Askar Tuganbaev, 2021. "On Rings of Weak Global Dimension at Most One," Mathematics, MDPI, vol. 9(21), pages 1-3, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2643-:d:660301
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    Cited by:

    1. Askar Tuganbaev, 2022. "Centrally Essential Rings and Semirings," Mathematics, MDPI, vol. 10(11), pages 1-74, May.

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