Proof of Theorem 8.7 in Atiyah-MacDonald

It seems to me that the second part of the proof of Theorem 8.7 p. 90 in Atiyah-MacDonald can be simplified. We must check the uniqueness of the decomposition of an Artin ring A as a finite product of Artin local rings A_i. To do this it suffices to observe that, for each minimal primary ideal q of A, there is a unique i such that q is the kernel of the canonical projection onto A_i.