On the uniform dimension of subextensions in skew polynomial rings

TL;DR

This paper explores the invariance of uniform dimension in subextensions of skew polynomial rings, extending classical results to non-commutative settings and identifying conditions for preservation of this dimension.

Contribution

It extends classical uniform dimension invariance results to skew polynomial rings, especially in non-commutative contexts, and introduces the concept of essentially special subextensions.

Findings

Uniform dimension invariance in skew polynomial subextensions.

Extension of classical polynomial ring results to skew Laurent polynomial rings.

Identification of conditions for preservation of uniform dimension in non-commutative subextensions.

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}}; \mathbf{\alpha} , \mathbf{\delta} ]$} of bijective type over a well-ordered set of variables. When the coefficient ring has enough uniform left ideals, in the commuting variables case we show that classical results on this topic for polynomial rings extend to subextensions of skew Laurent polynomial rings \mbox{$ \mathbb{S}=R[ \mathbf{\mathrm{X}}^{\pm1}; \mathbf{\alpha}]$}, 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.