A stable range for quadratic forms over commutative rings (Q1369608)

The quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results: (1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\). (2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\). They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above.