On commutative rings whose maximal ideals are idempotent

summary: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

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

summary:We provide some characterizations of rings $R$ for which every (finitely generated) module belonging to a class $\mathcal {C}$ of $R$-modules is a direct sum of cyclic submodules. We focus on the cases, where the class $\mathcal {C}$ is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules