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.