Abstract:We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.
Keywords: artinian module; modules of finite length; noetherian module; quasi-injective module; quasi-projective module; radical-fine class of modules; socle-fine class of modules
DOI: DOI 10.14712/1213-7243.2019.012
AMS Subject Classification: 13C13 13E05 13E10 13E99