Diophantine sets

Diophantine subsets of $\mathbb{Z}$ play a key role in the negative answer to Hilbert’s tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we prove that for every finitely presented scheme $Y$ over a ring $R$, there exists an affine $R$-scheme $X$ with a finitely presented $R$-morphism $X\to Y$ such that $X(R^{\mathrm{\prime }})\to Y(R^{\mathrm{\prime }})$ is surjective for every $R$-algebra $R^{\mathrm{\prime }}$.