2011/02/07 by Vered Moskowicz, Moskowicz, Vered
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA
paper · pdf · doi:10.48550/arxiv.1102.1416
arxiv created 2011/02/07 · openalex publication_date 2011/02/07 · arxiv updated 2011/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An almost PI algebra is a generalisation of a just infinite algebra which does not satisfy a polynomial identity. An almost PI algebra has some nice properties: It is prime, has a countable cofinal subset of ideals and when satisfying ACC(semiprimes), it has only countably many height 1 primes. Consider an affine prime Goldie non-simple non-PI k-algebra R of GK dimension <3, where k is an uncountable field. R is an almost PI algebra. We give some possible additional conditions which make such an algebra primitive. This gives a partial answer to Small's question: Let R be an affine prime Noetherian k-algebra of GK dimension 2, where k is any field. Does it follow that R is PI or primitive? We also show that the center of R is a finite dimensional field extension of k, and if, in addition, k is algebraically closed, then R is stably almost PI.