ON COMULTIPLICATION AND R-MULTIPLICATION MODULES

We state several conditions under which comultiplica-tion and weak comultiplication modules are cyclic and study strongcomultiplication modules and comultiplication rings. in particu-lar, we will show that every faithful weak comultiplication modulehaving a maximal submodule over a reduced ring with a nite in-decomposable decomposition is cyclic. also we show that if mis an strong comultiplication r-module, then r is semilocal andm is nitely cogenerated. furthermore, we dene an r-modulem to be p-comultiplication, if every nontrivial submodule of m isthe annihilator of some prime ideal of r containing the annihila-tor of m and give a characterization of all cyclic p-comultiplicationmodules. moreover, we prove that every p-comultiplication modulewhich is not cyclic, has no maximal submodule and its annihilatoris not prime. also we give an example of a module over a dedekinddomain which is not weak comultiplication, but all of whose local-izations at prime ideals are comultiplication and hence serves as acounterexample to [11, proposition 2.3] and [12, proposition 2.4].