The Golomb topology on Dedekind domains

The Golomb topology is the topology on the set of nonzero elements of a ring generated by the coprime cosets; the space obtained is called the Golomb space of , and is denoted by . When is a Dedekind domain with infinitely many ideals, this topology has many interesting properties: for example, is a Hausdorff space that is not regular, and is connected but locally disconnected at all of its points.
It is an open problem whether the Golomb topology on a Dedekind domain uniquely determines the algebraic properties of the domain, that is, if two nonisomorphic Dedekind domains can have homeomorphic Golomb spaces. Closely related to this question is the problem of determining all self-homeomorphisms of , and in particular if there is any self-homeomorphism that cannot be written as a composition of multiplication by units and automorphisms of the ring .
In this talk, I will show how several algebraic properties of a Dedekind domain can be obtained from its Golomb topology, obtaining partial results for the questions above. In particular, I will show that has no self-homeomorphism except for the identity and the multiplication by ; the method of the proof also allows to show that and are not homeomorphic for every number field . Moreover, I will analyze the context of polynomial rings, showing how to recognize several properties of from ; in particular, I will show that if are contained in the algebraic closure of for some prime number then and are not homeomorphic.