On the Linear Weak Topology and Dual Pairings over Rings

In this note we study the weak topology on paired modules over a (not necessarily commutative) ground ring. Over QF rings we are able to recover most of the well known properties of this topology in the case of commutative base fields. The properties of the linear weak topology and the dense pairings are then used to characterize pairings satisfying the so called $\alpha$-condition.