On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings

Let R be a commutative ring of dimension d, S=R[X] or R[X,1/X] and P a finitely generated projective S module of rank r. Then P is cancellative if P has a unimodular element and r≥d+1. Moreover if r≥dim⁡(S), then P has a unimodular element and therefore P is cancellative. As an application we prove that if R is a ring of dimension d of finite type over a Prüfer domain and P is a projective R[X] or R[X,1/X] module of rank at least d+1, then P has a unimodular element and is cancellative.