Dp-minimal integral domains.
Abstract: (joint with Yatir Halevi) As expected from the classification of dp-minimal fields, dp-minimal integral domains are close to be valuation domains, but not always. They are local, divided in the sense of Akiba, and every localisation at a non-maximal prime ideal is a valuation domain. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal. I will present these results as well as some examples of dp-minimal domains which are not valuation domains. If time allows it, I will also talk about a model-theoretic proof of a result of Echi and Khalfallah on the prime spectrum of the ring of bounded elements of the hyperreals.