Prime affine algebras of Gelfand-Kirillov dimension one (Q797666)

It is proved that any prime finitely generated algebra (over a field) which has Gelfand-Kirillov dimension one must be fully bounded noetherian, and must be a finitely generated module over its center. This contrasts with examples of \textit{R. S. Irving} and the first author of prime finitely generated algebras of finite GK-dimension which are not Goldie, and of non-prime finitely generated algebras of GK-dimension one which are not Goldie [Bull. Lond. Math. Soc. 15, 596-600 (1983; Zbl 0525.16009)]. It also contrasts with an example of \textit{J. C. McConnell} of a simple noetherian algebra of GK-dimension one which is not artinian [J. Algebra 76, 489-493 (1982; Zbl 0484.16013)].