Abstract
Let R be a commutative ring with identity. A unital R-module M is a comultiplication module provided that, for each submodule N of M, there exists an ideal A of R such that N is the set of elements m in M such that Am = 0. It is proved that every comultiplication module with zero radical is semisimple. Moreover, for any comultiplication module M, every submodule has a unique complement and a unique closure in M. Every Noetherian comultiplication module is an Artinian quasi-injective module. In case R is a semilocal ring containing precisely n distinct maximal ideals, for some positive integer n, every comultiplication R-module has Goldie dimension at most n. On the other hand, if R is a ring with finite Goldie dimension n, for some positive integer n, then it is proved that certain faithful comultiplication R-modules have hollow dimension at most n.