A dynamical comparison between the rings R⟨X⟩ and R(X)

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Afef Ellouz, Henri Lombardi and Ihsen Yengui
J. Algebra 320 , (2008), 521-533.

Abstract :
We prove that for any ring   R   with Krull dimension  ≤ k, the localization of   R[X]   at monic polynomials (denoted by  R⟨X⟩) "dynamically behaves" like the ring  R(X)  (i.e. the localization of   R[X]   at primitive polynomials) or a localization of a polynomial ring of type  RS[X]  where  S  is a multiplicatively closed subset of  R  with  dim(RS)  ≤ k-1.
As application, we give a simple and constructive proof of Lequain-Simis Theorem which is an important variation of the Quillen-Suslin Theorem. Our proof is based on a contructive variant of the Lequain-Simis Induction.