 
Summary: On the Linear Weak Topology and Dual Pairings over
Rings
Jawad Y. Abuhlail
Department of Mathematical Sciences
King Fahd University of Petroleum & Minerals
31261 Dhahran  Saudi Arabia
abuhlail@kfupm.edu.sa
Abstract
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 proper
ties of the linear weak topology and the dense pairings are then used to characterize
pairings satisfying the so called condition.
Introduction
Let R be a commutative field, V, W be vector spaces over R with a nondegenerating R
bilinear form : V ŚW R, P := (V, W) be the induced Rpairing and consider V
P
W
and W
P
