A criterion for rings which are locally valuation rings
Volume 116 / 2009
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.
