Extended Centres of Finitely Generated Prime Algebras

Bell, Jason P.; Smoktunowicz, Agata

doi:10.1080/00927870802254850

Cited by 4 publications

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References 13 publications

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“…Hence R is an almost PI algebra. Use Proposition 1.44 or Proposition 1.45 to conclude that C(R) is algebraic over k. Theorem 2.12 is a special case of a result of Bell and Smoktunowicz brought in [5]: "Let k be a field and let A be a finitely generated prime k-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) − 2 over k".…”

Section: 4mentioning

confidence: 99%

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

Moskowicz

2011

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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.

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Section: 4mentioning

confidence: 99%

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

Moskowicz

2011

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Growth, entropy and commutativity of algebras satisfying prescribed relations

Smoktunowicz

2014

Sel. Math. New Ser.

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In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [33], [34]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8].

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