Corollary 4

Let A be a graded commutative ring of type $\mathbf{N}$ and let $(A_n)$ be its graduation. For A to be Noetherian, it is necessary and sufficient that $A_n$ be Noetherian and that A be ajinitely generated A,-algebra.

The condition is sufficient by Corollary 3. Conversely, suppose A is Noetherian; $m = \sum_{n \geq 1} A_n$ which is an ideal of A, is then finitely generated; then A is a finitely generated A,-algebra (§ 1, no. 2, Corollary to Proposition 1); on the other hand $A_n$, which is isomorphic to $A/m$, is a Noetherian ring.