We state several conditions under which comultiplica-tion and weak coMULTIPLICATION modules are cyclic and study strong coMULTIPLICATION modules and coMULTIPLICATION rings. In particu-lar, we will show that every faithful weak coMULTIPLICATION module having a maximal submodule over a reduced ring with a nite in-decomposable decomposition is cyclic. Also we show that if M is an strong coMULTIPLICATION R-module, then R is semilocal and M is nitely cogenerated. Furthermore, we de ne an R-module M to be p-coMULTIPLICATION, if every nontrivial submodule of M is the annihilator of some prime ideal of R containing the annihila-tor of M and give a characterization of all cyclic p-coMULTIPLICATION modules. Moreover, we prove that every p-coMULTIPLICATION module which is not cyclic, has no maximal submodule and its annihilator is not prime. Also we give an example of a module over a Dedekind domain which is not weak coMULTIPLICATION, but all of whose local-izations at prime ideals are coMULTIPLICATION and hence serves as a counterexample to [11, Proposition 2. 3] and [12, Proposition 2. 4].