Characterisation of PF rings by the Finite Topology on duals of R-Modules

In this paper we study the properties of the finite topology on the dual of a module over an arbitrary ring. We aim to give conditions when certain properties of the field case are can be still found here. Investigating the correspondence between the closed submodules of the dual $M^{\ast }$ of a module $M$ and the submodules of $M$, we prove some characterisations of PF rings: the up stated correspondence is an anti isomorphism of lattices iff $R$ is a PF ring.