Stable rank of down-up algebras

TL;DR

This paper studies the structure of finitely generated projective modules over down-up algebras, revealing the existence of non-free stably free ideals and calculating their stable rank.

Contribution

It demonstrates that all noetherian down-up algebras have non-free stably free right ideals and computes their stable rank using advanced algebraic theorems.

Findings

Existence of non-free, stably free right ideals in all noetherian down-up algebras

Calculation of stable rank for these algebras

Application of Stafford's Stable Range Theorem and Kmax dimension

Abstract

We investigate the behavior of finitely generated projective modules over a down-up algebra. Specifically, we show that every noetherian down-up algebra $A(\alpha,\beta,\gamma)$ has a non-free, stably free right ideal. Further, we compute the stable rank of these algebras using Stafford's Stable Range Theorem and Kmax dimension.