On An Application Of The Suslin Monic Polynomial Theorem (I)

TL;DR

This paper establishes a characterization of the Krull dimension of finitely generated modules over polynomial rings over division rings, linking it to torsion properties relative to subrings, and proposes a generalization of this theorem.

Contribution

It introduces a new theorem relating Krull dimension to torsion modules over polynomial rings and generalizes this result to broader contexts.

Findings

Krull dimension equals m if and only if the module is non-torsion over B_m and torsion over larger subrings.

Provides a criterion to determine module dimension based on torsion properties.

Proposes a generalization of the main theorem to wider settings.

Abstract

In this paper our main theorem states the following, Main Theorem : Let B denote the polynomial ring D[x1,.... ,xn] , in the commuting indeterminates x i over a division ring D . Let M be a finitely generated B-module . Let B m denote the polynomial subring of B , namely D[x1,.... ,xm] , in m , indeterminates , where m is an integer such that 0 ? m ? n , with B0 =D, and Bn =B . Then Krull dimension (M) is m , 0 ? m ? n , if and only if M is a non torsion B m module such that for any positive integer k , k > m , M is a torsion B k module . We then also state and announce a generalisation of the above theorem .