In this chapter we will take a closer look at what Diophantine generation and Diophantine equivalence tell us about Diophantine decidability and definability over countable rings. We have already touched on these questions in our introduction. There we talked about the relationship between Diophantine definitions and Diophantine undecidability. To make this discussion more precise over rings other than the ring of rational integers, we will need to determine what the analog of a recursive function (or, more informally, an algorithm) is over these rings. To formalize the notion of an algorithm over countable structures, one uses presentations. If it exists, a recursive presentation of a given field F is a homomorphism from F into a field whose elements are natural numbers. Under this homomorphism all the field operations of F are interpreted by restrictions of recursive functions and the image of F is a recursive set. (Here we remind the reader that Appendix A contains definitions of recursive functions and recursive sets, as well as a list of references.) Not all fields and rings have such presentations. A field or ring which has such a presentation is called recursive. However, as we will see below, this notion of a presentation is too “strong” for our purposes. Presentations which are more suitable for a discussion of Diophantine questions are called “weak presentations.” We describe these presentations in the following section. Finally, we note that most of this chapter is based on [95].