A constructive comparison of the rings R(X) and R〈X〉 and application to the Lequain–Simis Induction T

Abstract

We constructively prove that for any ring R with Krull dimension d, the ring R X locally behaves like the ring R(X) or a localization of a polynomial ring of type (S −1 R)[X] with S a multiplicative subset of R such that the Krull dimension of S −1 R is d − 1. 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.