Constructing (almost) rigid rings and a UFD having infinitely generated Derksen and Makar-Limanov invariant

TL;DR

The paper presents an example of a unique factorization domain with an infinitely generated Derksen invariant, demonstrating an 'almost rigid' ring where the Derksen and Makar-Limanov invariants coincide, and discusses techniques to establish such properties.

Contribution

It introduces a novel example of an almost rigid UFD with an infinitely generated Derksen invariant and generalizes Mason's abc-theorem for this context.

Findings

Constructed an almost rigid UFD with infinitely generated Derksen invariant.

Established techniques for proving (almost) rigidity of rings.

Generalized Mason's abc-theorem for use in ring analysis.

Abstract

An example is given of a UFD which has infinitely generated Derksen invariant. The ring is \textquotedblleft almost rigid\textquotedblright\ meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.