Geodesic
A criterion for rings which are locally valuation rings
Kamran Divaani-Aazar 1   ; Mohammad Ali Esmkhani 2   ; Massoud Tousi 3
1 Department 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
2 Department of Mathematics Zanjan University P.O. Box 45195-313, Zanjan, Iran
3 Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746, Tehran, Iran
Colloquium Mathematicum, Tome 116 (2009) no. 2, pp. 153-164 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/cm116-2-2
Keywords: using notion cyclically pure injective modules characterization rings which locally valuation rings established applications characterizations fer domains pure semisimple rings provided namely domain fer only three classes pure injective cyclically pure injective rd injective modules equal prove commutative ring pure semisimple only every r module cyclically pure injective

Kamran Divaani-Aazar  1   ; Mohammad Ali Esmkhani  2   ; Massoud Tousi  3

1 Department 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
2 Department of Mathematics Zanjan University P.O. Box 45195-313, Zanjan, Iran
3 Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746, Tehran, Iran 
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 locally valuation rings
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Kamran Divaani-Aazar; Mohammad Ali Esmkhani; Massoud Tousi. A criterion for rings which are
 locally valuation rings. Colloquium Mathematicum, Tome 116 (2009) no. 2, pp. 153-164. doi: 10.4064/cm116-2-2

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