Exercise 16, § 1

Let A be a Noetherian ring. For A to be isomorphic to a finite product of integral domains, it is necessary and sufficient that for every prime ideal p of A the local ring A, be an integral domain. (To see that the condition is sufficient, note first that it implies that that ring A is reduced; deduce that $\{0\} = \bigcap_i p_i$, where the $p_i$ ($1 \leq i \leq n$) are the minimal prime ideals of A. Show that, for $i \neq j$, of necessity $p_i + p_j = A$; for this, note that, if there existed a maximal ideal m containing $p_i + p_j$, the ring $A_m$ would not be an integral domain, using Corollary 3 to Proposition 2 of no. 1, and the Corollary to Proposition 5 of no. 2.)