Structure of Galois rings and the Gelfand-Kirillov Conjecture

Abstract: The theory of Galois rings and orders, introduced by Futorny and Ovsienko, has many interesting applications to the structure and representation theory of algebras. This paper focuses on ring theoretical properties of Galois rings. The main technique is based on the fact that our algebras are embedded in a nice way into fixed rings of skew group (or monoid) rings, and via a simple localization procedure many facts about our rings can be deduced from properties of the associated skew group rings. With this tool we obtain natural conditions for our rings to be Ore domains and (semi)prime Goldie rings. We also discuss various ring theoretical dimensions and analyze what can be said when we combine powerful theories of Galois rings and PI-rings. We use our methods to compute dimensions and establish structural properties of affine and double affine Hecke algebras, as well as spherical Coulomb branch algebras. We also verify the Gelfand-Kirillov conjecture for the later and for the spherical subalgebras of the DAHA.