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An almost PI algebra is a generalisation of a just infinite algebra whichdoes not satisfy a polynomial identity. An almost PI algebra has some niceproperties: It is prime, has a countable cofinal subset of ideals and whensatisfying ACC(semiprimes), it has only countably many height 1 primes.Consider an affine prime Goldie non-simple non-PI $k$-algebra $R$ of GKdimension $<3$, where $k$ is an uncountable field. $R$ is an almost PI algebra.We give some possible additional conditions which make such an algebraprimitive. This gives a partial answer to Small's question: Let $R$ be anaffine 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, $k$ isalgebraically closed, then $R$ is stably almost PI.

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