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Prime affine algebras of Gelfand-Kirillov dimension one. (English) Zbl 0545.16011
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 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 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)].
Reviewer: K.R.Goodearl

MSC:
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16P40 Noetherian rings and modules (associative rings and algebras)
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[1] \scG. Bergman, unpublished.
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[5] McConnell, J, Representations of solvable Lie algebras. V. on the Gelfand-Kirillov dimension of simple modules, J. algebra, 76, 489-493, (1982) · Zbl 0484.16013
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