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

研究非对易环和代数的基本结构性质，涉及到零代数、代数和代数的增长性(Gelfand-Kirillov维)等概念，在这三个领域都有非常困难的公开问题，并且这三个主题之间存在着许多相互联系。这个项目将研究其中的几个公开问题。在零代数领域最著名的问题是1930年首次提出的Koethe猜想，它断言如果一个环没有非零零理想，那么它就没有非零零单边理想。这是关于环的一般结构的一个基本问题，对零环和幂零环的透彻理解是任何认真尝试理解一般环所必需的。与零环相关的问题将是该项目的主要主题之一。这位学者已经在这个领域做出了基本的贡献，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，它询问一个有限生成的代数在基域上是代数的知识是否足以确保该代数是有限维的。正如戈洛德和沙法雷维奇在1964年所论证的那样，这在总体上是不正确的。然而，许多部分积极的结果是已知的，该项目的第二个主要主题是澄清Kurosh问题的积极和消极解决方案之间的边界。这个主题和前面的主题有密切的联系：例如，Golod-Shafarevich代数是非幂零的无限维零代数。第三个主题是代数的增长，特别是对增长受限的代数的研究。这位研究员在证明Artin-Stafford Gap定理方面已经对这一领域做出了根本贡献，该定理断言没有严格在2到3之间增长的分次域。第三个主题的很大一部分将是在限制代数增长的情况下研究前两个主题中出现的问题。例如，Golod-Shafarevich代数有指数增长，但这个家伙最近给出了一个(相对)小增长的例子。在该项目中，将对许多公开问题中的生长条件的确切限制进行调查。

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.