Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

TL;DR

This paper extends the cancellation theorem to noncommutative domains of Gelfand-Kirillov dimension one over fields of characteristic zero, and provides counterexamples in positive characteristic, also exploring skew polynomial extensions.

Contribution

It introduces noncommutative analogues of the Abhyankar-Eakin-Heinzer cancellation theorem and establishes conditions for cancellativity in this broader setting.

Findings

Noncommutative Gelfand-Kirillov dimension one domains are cancellative in characteristic zero.

Counterexamples show non-cancellativity in positive characteristic.

A skew analogue of the cancellation theorem is proved.

Abstract

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.