ON COMULTIPLICATION AND R-MULTIPLICATION MODULES

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].