Abstract:

Abstract: The most famous open problem in the area of nil algebras is the Köthe 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. The most famous problem about algebraic algebras is the Kurosh Problem, which is of a similar vintage and 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 the borderline between positive and negative solutions of the Kurosh Problem is still being investigated. There are close connections between these two general themes; for example, the Golod-Shafarevich algebras are infinite dimensional finitely generated nil algebras that are not nilpotent. My talk will be a short survey of the current state of these two themes, including more on their relationship.