Proposition 9

Let A be a Noetherian ring. The following conditions are equivalent : (a) A is Artinian. (b) All the prime ideals $\mathfrak{a}$ in A are maximal ideals. (c) All the elements of Ass(A) are maximal ideals.

If these conditions are fulfilled, A has only a finite number of prime ideals, which are all maximal and associated with the A-module A; further, A is a semi-local ring and its Jacobson radical is nilpotent.

To say that A is Artinian is equivalent to saying that A is an A-module of finite length; hence (a) and (c) are equivalent by Proposition 7. Clearly (b) implies (c). Finally, (a) implies (b) since every Artinian integral domain is a field. The properties (a), (b) and (c) are therefore equivalent.

Suppose they hold. As every prime ideal of A belongs to Supp(A) and every element of Supp(A) contains an element of Ass(A) ($\S$ 1, no. 3, Proposition 7), it follows from (c) that Ass(A) is the set of all prime ideals of A; then A has only a finite number of prime ideals, all of them maximal and associated with the A-module A. This obviously implies that A is semi-local; finally, we know that the Jacobson radical of an Artinian ring is nilpotent (Algebra, Chapter VIII, § 6, no. 4, Theorem 3).