arXiv subjects
Publications and source records attributed to Be'eri Greenfeld.
We realize any submultiplicative increasing function which is equivalent to a polynomial proportion of itself as the growth function of a finitely generated simple Lie algebra. As an application, we resolve two open problems posed by Petrogradsky on Lie algebras with intermediate growth.
We construct an increasing, submultiplicative, arbitrarily rapid function which is not equivalent to the growth function of any finitely generated algebra, demonstrating the difficulty in characterizing growth functions in an asymptotic language.
We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Nielsen for algebras over infinite fields. If $R$ is a graded algebra generated in degree $1$ over a field of characteristic zero and $δ$ is a grading preserving derivation on $R$, then the Jacobson radical of $R$ is $δ$-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring $R[[X;σ,δ]]$ is well-defined whenever $δ$ is a locally nilpotent $σ$-derivation; this answers a conjecture by Bergen and Grzeszczuk and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on $δ$.
We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated: We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated, and on the other hand construct a finitely generated, infinite dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.
Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regarding identifying von Neumann regular rings via their prime factors holds for Leavitt path algebras.
Araújo, Kinyon and Konieczny (2011) pose several problems concerning the construction of arbitrary commuting graphs of semigroups. We observe that every star-free graph is the commuting graph of some semigroup. Consequently, we suggest modifications for some of the original problems, generalized to the context of families of semigroups with a bounded number of generators, and pose related problems. We construct monomial semigroups with a bounded number of generators, whose commuting graphs have an arbitrary clique number. In contrast to that, we show that the diameter of the commuting graphs of semigroups in a wider class (containing the class of nilpotent semigroups), is bounded by the minimal number of generators plus two. We also address a problem concerning knit degree.
We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent paper. Our construction goes through a construction of finitely generated just-infinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type. We also discuss the connection between entropy of algebras and their homomorphic images, as well as the degrees of their generators of free subalgebras.
Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.
Bartholdi and Smoktunowicz constructed 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 with many regular enough growth types. Then, inspired by an example by Zelmanov, 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.
We construct affine algebras with an arbitrary amount of simple modules of each dimension.
The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of subchains in a chain of primes in a PI-algebra.