The prime spectrum of algebras of quadratic growth

JOURNAL OF ALGEBRA

卷:319

Article

Bell, Jason P.²  Smoktunowicz, Agata¹ 

[1] Univ Edinburgh, Sch Math, Maxwell Inst Sci, Edinburgh EH9 3JZ, Midlothian, Scotland

[2] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada

关键词: GK dimension;    quadratic growth;    primitive rings;    PI rings;    graded algebra;   

DOI  :  10.1016/j.jalgebra.2007.08.026

来源: Elsevier

【 摘 要 】

We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that A/P has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that A I P has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that A/P has GK dimension 1. (C) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free