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

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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.