Exercise 12, § 8

We suppose that the ring A is integral. Now prove that the following conditions are equivalent: α) $\mathrm{dh}(A) \leq 1$. β) Every divisible a-module is injective. γ) The ring A is prüferian (exercise 11) and noetherian. (To show that α) implies β), use exercise 10.) One then says that A is a Dedekind ring.