One of the most important rings we study is the ring of integers. It was our first example of an algebraic structure: the first polynomial ring that we examined was \({\mathbb Z}[x]\text{.}\) We also know that the integers sit naturally inside the field of rational numbers, \({\mathbb Q}\text{.}\) The ring of integers is the model for all integral domains. In this chapter we will examine integral domains in general, answering questions about the ideal structure of integral domains, polynomial rings over integral domains, and whether or not an integral domain can be embedded in a field.
