On the uniform dimension of subextensions in skew polynomial rings

TL;DR

This work investigates when the left uniform dimension $\mathsf{udim}$ is preserved under subextensions of skew polynomial rings $S=R[ \boldsymbol{X}; \boldsymbol{\alpha}, \boldsymbol{\delta} ]$ of bijective type. It develops a general framework based on the notion of enough uniform submodules and nicely essential extensions, providing lifting techniques that transfer $\mathsf{udim}$ and left Goldie properties from the coefficient ring $R$ to subextensions $A$. In the commuting-variables setting, it solves subextension problems for skew Laurent polynomial rings, establishing invariance theorems and Goldie-ness results; in the noncommuting case, it identifies essential special subextensions where invariance still holds, while highlighting the inherent complexity and potential pathologies. The paper also presents easy examples showing the limits of Shock-type arguments in the noncommuting context and demonstrates a broad array of essentially special subextensions, including two-variable constructions, where the uniform dimension remains anchored to that of the base ring $R$ under suitable hypotheses.

Abstract

This work investigates the invariance of the non-necessarily finite uniform dimension and related concepts for subextensions in skew polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}; \mathbfα , \mathbfδ ]$} of bijective type over a well-ordered set of variables. When the coefficient ring has enough uniform left ideals, in the commuting variables case we show that classical results on this topic for polynomial rings extend to subextensions of skew Laurent polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}^{\pm1}; \mathbfα]$}, generated over $R$ by any family of (standard) terms. The situation in the non-commuting variables context is more complex; easily formed polynomial-like subrings can behave very oddly from the ambient ring. We provide easy examples of a (semi)prime left Goldie skew polynomial ring of bijective type containing a monoid subring isomorphic to a free non-commutative polynomial ring. We then study the so-called subclass of \emph{essentially special subextensions} and obtain for them the preservation of the uniform dimension and related concepts.

Theorems & Definitions (50)

• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof