Prime and primitive algebras with prescribed growth types (Q2014282)

In [Q. J. Math. 65, No. 2, 421--438 (2014; Zbl 1312.16017)], \textit{A. Smoktunowicz} and \textit{L. Bartholdi} constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. The authors showed that their construction does not necessarily result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type. Using a construction of an inverse system of monomial ideals which arise from this construction, the author presents the construction of finitely generated primitive algebras without further limitations on the growth type. The author is inspired by \textit{E. I. Zel'manov}'s example in [Sib. Math. J. 20, 303--304 (1979; Zbl 0433.16005)], so in this paper, he studies how prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of primitive constructions. In fact, the reader will find good results about this area of algebras, where the author discusses deeply this problem, and explains some results by examples about non-semiprime algebras. Moreover, the author closes this paper with two open questions, which are: What are the possible growth types of finitely generated simple algebras? What are the possible growth types of finitely generated domain?