Topic
Kurosh problem
About: Kurosh problem is a research topic. Over the lifetime, 14 publications have been published within this topic receiving 91 citations.
Papers
1 Jan 1984
TL;DR: In this article, the authors discuss several questions, which loosely can be stated as follows: does this or that free object appear in a skew field which satisfies certain conditions? Free objects which they have in mind are: free semigroup, free group and free algebra with two generators.
Abstract: In my talk I am going to discuss several questions, which loosely can be stated as follows: does this or that free object appear in a skew field which satisfies certain conditions? Free objects which I have in mind are: free semigroup, free group and free algebra with two generators. It is not reasonable to consider a bigger number of generators, because as is well known, every free object mentioned contains the corresponding free object on a countable number of generators. On the other hand in the case of one generator every object is commutative, and these questions have been considered. It is well known that every skew field which does not coincide with its center contains a free subgroup with one generator (in its multiplicative group). The question about one-generator sub- algebras constitutes the famous Kurosh problem which is still very far from the solution. However, it seems more appropriate to me to consider noncommutative subobjects in the skew field setting.
31 citations
TL;DR: In this paper, it was shown that any GI-ring is Dedekind finite (von Neumann finite) and Nilpotent elements of a semiprimitive GI ring have bounded index.
19 citations
TL;DR: In this article, the authors consider the problem of whether there is an infinite-dimensional algebraic algebra for linear multi-operator algebras over a field and show that, given an arbitrary signature, there is a variety of algebraic varieties of this signature such that the free algebra of the variety contains polylinear elements of arbitrarily large degree, while the clone of every such element satisfies some nontrivial identity.
Abstract: We consider a couple of versions of the classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains polylinear elements of arbitrarily large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then each such clone may be assumed to be finite-dimensional. Our approach is the following: we cast the problem in the language of operads and then apply the usual homological constructions in order to adopt Golod’s solution to the original Kurosh problem. This paper is expository, so that some proofs are omitted. At the same time, the general relations of operads, algebras, and varieties are widely discussed.
11 citations
TL;DR: In this paper, it was shown that for any field k and any monotonically increasing function f(n) which grows super-polynomially but subexponentially, there exists an infinite-dimensional finitely generated nil k-algebra whose growth is asymptotically bounded by f (n).
10 citations
Posted Content•
TL;DR: In this paper, the authors consider the problem of whether there is an infinite-dimensional algebraic algebra for linear multi-operator algebras over a field, and they show that, given an arbitrary signature, there is a variety of algebraic varieties of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity.
Abstract: We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional.
Our approach is the following: we translate the problem to the language of operads and then apply usual homological constructions, in order to adopt Golod's solution of the original Kurosh problem.
The paper is expository, so that some proofs are omited. At the same time, the general relations of operads, algebras, and varieties are widely discussed.
8 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 1 |
| 2019 | 1 |
| 2015 | 2 |
| 2011 | 2 |
| 2009 | 2 |