Definition 1

A Krull domain all of whose non-zero prime ideals are maximal is called a Dedekind domain.

Examples of Dedekind domains (1) Every principal ideal domain is a Dedekind domain. (2) Let K be a finite extension of $\mathbf{Q}$ and A the integral closure of $\mathbf{Z}$ in K. The ring A is a Krull domain (§ 1, no. 8, Proposition 12). Let $\mathfrak{p}$ be a non-zero prime ideal of A. Then $\mathfrak{p} \cap \mathbf{Z}$ is non-zero (Chapter V, § 2, no. 1, Corollary to Proposition 1) and hence is a maximal ideal of $\mathbf{Z}$; hence $\mathfrak{p}$ is a maximal ideal of A (loc. cit., Proposition 1). Therefore, A is a Dedekind domain. In general, A is not a principal ideal domain (Algebra, Chapter VII, § 1, Exercise 12). (3) \* Let V be an affine algebraic variety and A the ring of functions regular on V. Suppose that A is not a field (i.e. that V is not reduced to a point). For A to be a Dedekind domain, it is necessary and sufficient that V be an irreducible curve with no singular point: for to say that A is an integral domain amounts to saying that V is irreducible; to say that every non-zero prime ideal of A is maximal amounts to saying that A is a curve; finally, as A is Noetherian, to say that it is a Krull domain amounts to saying that it is integrally closed, that is that V is a normal curve, or also that it has no singular point. \* (4) A ring of fractions $S^{-1}A$ of a Dedekind domain A is a Dedekind domain if $0 \notin S$. For $S^{-1}A$ is a Krull domain (§ 1, no. 4, Proposition 6) and every non-zero prime ideal of $S^{-1}A$ is maximal by Chapter II, § 2, no. 5, Proposition 11.