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Recall that an R-module M is pure-semisimple if every module in the category σ [ M ] is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that "a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring". Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is "whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?" The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals M = R x ⊕ T which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. [ABSTRACT FROM AUTHOR]

We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules 0 ≠ K , L ⊆ M , the factor modules K / Rad (K) , and L / Rad (L) are virtually simple and isomorphic (an R-module M is virtually simple if, M ≠ 0 and M ≅ N for every nonzero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial. [ABSTRACT FROM AUTHOR]

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 leftR-module is projective if and only if every FC-pure projective leftR-module is injective, if and only if,Ris 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 leftR-module is FC-pure projective if and only if every indecomposable leftR-module is a finitely presented cyclicR-module, if and only if,Ris 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 ringRis quasi-Frobenius if and only ifRis an Artinian ring and I-pure injective, if and only if,Ris an Artinian ring and the injective envelopeE(R) is an FC-pure projectiveR-module. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ringRis left Noetherian if and only if every C-pure projective leftR-module is pure projective. Also, over a left hereditary Noetherian ringR, a leftR-moduleMis C-pure projective if and only ifM = N⊕P, whereNis a direct sum of cyclic modules andPis a projective leftR-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that ifRis a local duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis a principal ideal ring. IfRis a left perfect duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis left Köthe (i.e., every leftR-module is a direct sum of cyclic modules). Also, it is shown that for a ringR, if every C-pure projective leftR-module is RD-projective, thenRis left Noetherian, every p-injective leftR-module is injective and every p-flat rightR-module is flat. Finally, it is shown that ifRis a left p.p-ring and every C-pure projective leftR-module is RD-projective, thenRis left Noetherian hereditary. The converse is also true whenRis commutative, but it does not hold in the noncommutative case. [ABSTRACT FROM PUBLISHER]