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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm116-2-2,
author = {Kamran Divaani-Aazar and Mohammad Ali Esmkhani and Massoud Tousi},
title = {A criterion for rings which are locally valuation rings},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {153-164},
zbl = {1177.13047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm116-2-2}
}
Kamran Divaani-Aazar; Mohammad Ali Esmkhani; Massoud Tousi. A criterion for rings which are locally valuation rings. Colloquium Mathematicae, Tome 116 (2009) pp. 153-164. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm116-2-2/