Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions.∗

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz’ “weakly compressible” modules. In particular we are interested in representing weakly compressible modules as a subdirect product of “prime” modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.