Remark (2)

The conditions of Proposition 9 for a Noetherian ring A imply that the spectrum of A is finite and discrete, every point of Spec(A) being therefore closed (Chapter 11, § 4, no. 3, Corollary 6 to Proposition 11). Conversely, for a Noetherian ring A, to say that every point $\mathfrak{a}$ of Spec(A) is closed means that every prime ideal of A is maximal (loc. cit..) and hence this condition is equivalent to that of Proposition 9.