Exercise 10, § 2

Let $A$ be a ring in which every ideal is a product of prime ideals. (a) Show that, for every prime ideal $p$ of $A$, $A/p$ is a Dedekind domain (Exercise 9). (b) Show that the set of minimal prime ideals $p_i$ of $A$ is finite (decompose $O$ as a product of prime ideals). (c) Supposethat $A/p_i$ is not a field. Show that, if $y \in p_i$, then, for all $x \in A - p_i$, there exists $z \in p_i$ such that $y = zx$. (Consider in $A$ the decompositions as products of prime ideals of $Ax$ and $Ax + Ay$ and the corresponding decompositions in $A/p_i$.) Deduce that, for all non-zero $y \in p_i$, $Ay = p_i$ (consider $p_i/Ay$, decomposing $Ay$ as a product of prime ideals); show finally that $p_i = p_i^2$ and therefore that there exists in $p_i$ an idempotent $e_i$ such that $p_i = Ae_i$ (cf. Chapter II, § 4, Exercise 15). Then $A$ is the direct composition of the ring $Ae$, and the ring $A(1 - e_i)$, isomorphic to the Dedekind domain $A/p_i$. (d) Suppose now that all the $p_i$ are maximal, so that $A$ is a direct composition of rings of the form $A/p_i^{n_i}$ (Chapter II, § 1, no, 2, Proposition 5) and we may therefore confine our attention to the case where $A$ is primary or where the unique prime ideal $p$ of $A$ is nilpotent. Show that in this case the hypothesis implies that $A$ is a principal ideal ring (Algebra, Chapter VII, § 1, Exercises 5 and 6). (e) Conclude from (c) and (d) that $A$ is the product of a finite number of Dedekind domains and a principal ideal ring.