Inversion of monic polynomials and existence of unimodular elements. II (Q1099224)

[For part I see the author's joint paper with \textit{A. Roy} [ibid. 183, 87-94 (1983; Zbl 0495.13007).] M. Roitman has asked the following question: Let \(R\) be a commutative noetherian ring and let \(P\) be a projective \(R[T]\)-module. Does \(P\) contain a unimodular element if \(P_ f\) contains a unimodular element for some monic polynomial \(f(T)\in R[T]?\) We prove that the answer to the question is affirmative in the following two cases: (1) \(\text{rank}(P)=2\); (2) \(R\) is a normal affine domain over an algebraically closed field and \(\text{rank}(P)=\text{dim}(R)\).