We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative R-algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring R is not unital, we give necessary and sufficient conditions for R to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring R^, its definable unitazation.
Annalisa Conversano (Mon,) studied this question.