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) (this happens, for example, when it is affine), 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, see Theorems 2.6, 2.7, 2.9 and 2.10. 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 (Theorem 2.11), and if, in addition, k is algebraically closed, then R is stably almost PI (Theorem 2.13).I m = I m+1 = I m+2 = . . .. So I m ∈ T is an upper bound for our chain. Hence Zorn's Lemma implies that T contains a maximal element, call it I. R/I is an almost PI algebra; Indeed, I ∈ T implies that R/I is non-PI, and maximality of I implies that for every I J, R/J is PI, so (R/I)/(J/I) ∼ = R/J is PI.Therefore, from Theorem 1.1, R/I is prime. Hence I is a (nonzero) prime ideal of R, so by assumption R/I is PI. We reached a contradiction, so there exists no such nonzero ideal Ĩ with R/ Ĩ non-PI.Of course, a similar criterion can be given for a just infinite algebra. Observe that a just infinite algebra A is actually weakly Noetherian. Indeed, let I 1 I 2 I 3 . . . be an ascending chain of proper ideals in A.Proposition 1.5. Let R be a weakly Noetherian infinite dimensional k-algebra. Assume that for every nonzero prime ideal P , R/P is finite dimensional over k. Then R is a just infinite algebra.Proof. Just replace in the proof of Corollary 1.4: non-PI by infinite dimensional, PI by finite dimensional, almost PI by just infinite, and Theorem 1.1 by [7, Theorem 1] (which says that a just infinite algebra is prime, whether affine or not).Next, we mimick an observation of Small ([17, Proposition 3.2] or [8, Lemma 2.1]), but again instead of affinity we assume ACC(ideals).Lemma 1.6. Let R be a weakly Noetherian non-PI k-algebra. Then R has a homomorphic image R which is prime and almost PI.