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) |
Keywords:
prime finitely generated algebra; Gelfand-Kirillov dimension; fully bounded noetherian; center; simple noetherian algebraReferences:
| [2] | Borho, W., On the Joseph-Small additivity principle for Goldie ranks, Comput. Math., 47, 3-29 (1982) · Zbl 0502.17007 |
| [3] | Chatters, A.; Hajarnavis, C., Rings with Chain Conditions (1980), Pitman: Pitman London · Zbl 0446.16001 |
| [4] | Irving, R.; Small, L. W., The Goldie conditions for algebras of bounded growth, Bull. London Math. Soc., 15, 596-600 (1983) · Zbl 0525.16009 |
| [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 |
| [6] | Resco, R., Transcendental division algebras and simple Noetherian rings, Israel J. Math., 32, 236-256 (1979) · Zbl 0404.16012 |
| [7] | Schelter, W., On the Krull-Akizuki theorem, J. London Math. Soc. (2), 13, 263-264 (1976) · Zbl 0331.16017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
