A+ CATEGORY SCIENTIFIC UNIT

A criterion for rings which are locally valuation rings

Volume 116 / 2009

Kamran Divaani-Aazar, Mohammad Ali Esmkhani, Massoud Tousi Colloquium Mathematicum 116 (2009), 153-164 MSC: 13F05, 13F30. DOI: 10.4064/cm116-2-2

Abstract

Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain $R$ is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring $R$ is pure semisimple if and only if every $R$-module is cyclically pure injective.

Authors

  • Kamran Divaani-AazarDepartment of Mathematics
    Az-Zahra University
    Vanak, Post Code 19834
    Tehran, Iran
    and
    Institute for Studies
    in Theoretical Physics and Mathematics
    P.O. Box 19395-5746, Tehran, Iran
    e-mail
  • Mohammad Ali EsmkhaniDepartment of Mathematics
    Zanjan University
    P.O. Box 45195-313, Zanjan, Iran
    e-mail
  • Massoud TousiInstitute for Studies
    in Theoretical Physics and Mathematics
    P.O. Box 19395-5746, Tehran, Iran
    e-mail

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