Abstract
Bartholdi and Smoktunowicz constructed in 2014 finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type.
Moreover, using a construction of an inverse system of monomial ideals which arise from this construction, we are able to further construct finitely generated primitive algebras without further limitations on the growth type.
Then, inspired by Zelmanov’s example in 1979, we show how our prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of the primitive constructions.
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The author wishes to thank Prof. Agata Smoktunowicz and Prof. Rostislav Grigorchuk for interesting related correspondence, and Prof. Uzi Vishne and Prof. Louis Rowen for their remarks about the paper. The author thanks the referee for his/her comment.
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Greenfeld, B. Prime and primitive algebras with prescribed growth types. Isr. J. Math. 220, 161–174 (2017). https://doi.org/10.1007/s11856-017-1513-z
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DOI: https://doi.org/10.1007/s11856-017-1513-z