In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $\pi ⁎:Ho{m}_R(G,G)\cong Ho{m}_R(G,H)$, where π⁎(φ) = πφ for each $\phi \in Ho{m}_R(G,G)$ (where maps are acting on the left). On the one hand,...

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

First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if $R$ is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication $R$-modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.