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Abstract: We prove constructively that for any ring R of Krull dimension ⩽1 and , the group acts transitively on . In particular, we obtain that for any ring R with Krull dimension ⩽1, all finitely generated stably free modules over are free. This settles the long-standing Hermite ring conjecture for rings of Krull dimension ⩽1. [Copyright &y& Elsevier]

Abstract: We constructively prove that for any ring R with Krull dimension ⩽d, the ring locally behaves like the ring or a localization of a polynomial ring of type with S a multiplicative subset of R such that the Krull dimension of is . As an application, we give a simple and constructive proof of the Lequain–Simis Induction Theorem which is an important variation of the Quillen Induction Theorem. [Copyright &y& Elsevier]