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

TL;DR

This paper investigates the properties of the finite topology on dual modules over arbitrary rings, establishing conditions under which properties from the field case hold and characterizing PF rings via lattice isomorphisms.

Contribution

It provides new characterizations of PF rings by analyzing the lattice correspondence between submodules of a module and its dual under the finite topology.

Findings

The lattice of closed submodules of the dual is anti-isomorphic to the submodules of the original module iff R is a PF ring.

Conditions under which properties of the field case extend to arbitrary rings are identified.

Characterizations of PF rings are established through lattice-theoretic properties.

Abstract

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 $M^{*}$ 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.