The Hermite ring conjecture in dimension one (Q947534)

Here is the author's abstract: ``We prove constructively that for any ring \textbf{R} of Krull dimension \(\leq1\) and \(n\geq3\), the group E\(_n({\mathbf R}[X])\) acts transitively on Um\(_n({\mathbf R}[X])\). In particular, we obtain that for any ring \textbf{R} with Krull dimension \(\leq 1\), all finitely generated stably free modules over \textbf{R}\([X]\) are free. This settles the long-standing Hermite ring conjecture for rings of Krull dimension \(\leq1\).'' In view of these results, the author conjectures that polynomial extensions in any number of indeterminates of a ring of Krull dimension \(\leq1\) are Hermite rings. The rings in this paper are commutative.