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# 38 - M. Behboodi (Joint with A. Ghorbani, A. Moradzadeh-Dehkordi, and S.H. Shojaee)و On FC-purity and I-purity of modules and Kothe rings, Comm. Algebra 42 (2014), 2061-2081.(ISI)

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a ﬁnitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure ﬂatness and I-pure ﬂatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.