Nil algebras, algebraic algebras and algebras with finite Gelfand-Kirillov dimension.

本文拟对非交换环和代数的基本结构性质进行研究，涉及零代数、代数代数和代数生长（Gelfand-Kirillov维数）的概念。这两个领域都有非常困难的开放性问题，而且这三个主题之间也有许多相互联系。该项目将调查其中几个悬而未决的问题。零代数领域中最著名的问题是1930年首次提出的Koethe猜想，该猜想断言，如果一个环没有非零零零理想，那么它就没有非零零零单侧理想。这是一个关于环一般结构的基本问题，对于任何认真理解环的尝试来说，彻底理解零环和幂零环是必要的。有关零环的相关问题将是本项目的主题之一。该研究员已经在这一领域做出了根本性的贡献，包括在任何可数域上构造一个简单的零代数。关于代数最著名的问题是Kurosh问题，这个问题是关于一个有限生成的代数是基域上的代数的知识是否足以保证该代数是有限维的。这通常是不正确的，正如Golod和Shafarevich在1964年所证明的那样。然而，许多部分的积极结果是已知的，该项目的第二个主题是澄清黑什问题的积极和消极解决方案之间的界限。这个主题和之前的主题有密切的联系：例如，gold - shafarevich代数是无限维的零代数，它们不是幂零的。第三个主题是代数的生长，特别是对限制生长代数的研究。这位研究员已经在这个领域做出了根本性的贡献，他证明了Artin-Stafford Gap定理，该定理断言没有严格在2和3之间增长的分级域。第三主题的实质性部分将是在代数增长的限制下研究前两个主题中出现的问题。例如，gold - shafarevich代数具有指数增长，但研究员最近给出了一个（相对）小增长的例子。该项目将研究许多开放问题中生长条件的确切限制。

It is proposed to undertake a study of fundamental structural properties of noncommutative rings and algebras involving the notions of nil algebras, algebraic algebras and growth of algebras (Gelfand-Kirillov dimension).There are very difficult open problems in each of these areas, and also there are many interconnections between the three main themes. The project will investigate several of these open problems.The most famous problem in the area of nil algebras is the Koethe Conjecture, first posed in 1930, which asserts that if a ring has no nonzero nil ideals then it has no nonzero nil one-sided ideals. This is a fundamental question about the general structure of rings, and a thorough understanding of nil and nilpotent rings is necessary for any serious attempt to understand general rings. Related problems concerning nil rings will be one of the main themes of this project. The Fellow has already made fundamental contributions to this area, including the construction of a simple nil algebra over any countable field.The most famous problem about algebraic algebras is the Kurosh Problem which asks whether the knowledge that a finitely generated algebra is algebraic over a base field is sufficient to ensure that the algebra is finite dimensional. This is untrue in general, as demonstated by Golod and Shafarevich in 1964. However, many partial positive results are known, and a second main theme of the project is to clarify the borderline between positive and negative solutions of the Kurosh Problem. There are close connections between this theme and the previous theme: for example, the Golod-Shafarevich algebras are infinite dimensional nil algebras that are not nilpotent.The third main theme is the growth of algebras, and, in particular, a study of algebras with restricted growth. The Fellow has already made a fundamental contribution to this area in proving the Artin-Stafford Gap Theorem, which asserts that there are no graded domains with growth strictly between 2 and 3.A substantial part of the third theme will be to investigate the problems arising in the first two themes under restrictions on the growth of the algebras. For example, the Golod-Shafarevich algebra has exponential growth, but the Fellow has recently produced a examples with (relatively) small growth. The exact limits on the growth conditions in many of the open problems will be investigated in the project.