Prime affine algebras of GK dimension two which are almost PI algebras

Details Prime affine algebras of GK dimension two which are almost PI algebras Prime affine algebras of GK dimension two which are almost PI algebras

2011-02-07

arxiv.org/abs/1102.1416v1

Mathematics

Rings and Algebras

Scientific paper

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.

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