On QF rings and artinian principal ideal rings

Authors : Alejandro ALVARADO-GARCÍA, César CEJUDO-CASTİLLA, Hugo Alberto RİNCÓN-MEJÍA, İvan Fernando VİLCHİS-MONTALVO, Manuel Gerardo ZORRİLLA-NORİEGA

Pages : 67-74

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Publication Date : 2019-02-01

Article Type : Research Paper

Abstract :In this work we give sufficient conditions for a ring $R$ to be quasi-Frobenius, such as $R$ being left artinian and the class of injective cogenerators of $R$-Mod being closed under projective covers. We prove that $R$ is a division ring if and only if $R$ is a domain and the class of left free $R$-modules is closed under injective hulls. We obtain some characterizations of artinian principal ideal rings. We characterize the rings for which left cyclic modules coincide with left cocyclic $R$-modules. Finally, we obtain characterizations of left artinian and left coartinian rings.
Keywords : left artinian ring, artinian principal ideal ring, conoetherian ring, coartinian ring, QF ring, perfect ring, semiartinian ring