Corollary

The prime ideals of an Artinian commutative ring are its maximal ideals.

In any commutative ring, a maximal ideal is prime. Let A be an Artinian commutative ring. Let $\mathfrak{p}$ be a prime ideal of A. The ring $A/\mathfrak{p}$ is an integral domain and is Artinian (Proposition 5), hence is a field (VIII, p. 5, Example 3). Consequently, the ideal $\mathfrak{p}$ is maximal.

The polynomial ring $\mathbf{Q}[(X_n)_{n\in\mathbf{N}}]$ is an integral domain; it is not Noetherian (or Artinian) (VIII, p. 15, Exercise 9). It is a subring of its field of fractions, which is an Artinian (and Noetherian) ring.