  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.
