On QF rings and artinian principal ideal rings

Year 2019, Volume: 48 Issue: 1, 67 - 74, 01.02.2019

Alejandro Alvarado-garcía César Cejudo-castilla Hugo Alberto Rincón-mejía İvan Fernando Vilchis-montalvo Manuel Gerardo Zorrilla-noriega

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

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