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I have a problem with one step of the proof of the 1st uniqueness theorem of a primary decomposition in Atiyah, MacDonald Commutative Algebra.

Theorem 4.5. Let $\mathfrak{a}$ be a decomposable ideal in the ring $A$ and $\mathfrak{a}=\bigcap_{1\leq i\leq n}\mathfrak{q}_i$ be a minimal primary decomposition of $\mathfrak{a}$. Let $\mathfrak{p}_i=\mathrm{rad}(\mathfrak{q}_i)$, ${1\leq i\leq n}$. Then the $\mathfrak{p}_i$ are precisely the prime ideals which occur in the set of ideals $\mathrm{rad}(\mathfrak{a}:x)$, $x\in A$, and hence are independent of the particular decomposition of $\mathfrak{a}$.

I can see that the statement is true for $x\notin \mathfrak{a}$, but for the case $x\in \mathfrak{a}$, $(\mathfrak{a}:x)=(1)$ and so $\mathrm{rad}(\mathfrak{a}:x)=(1)$? What am I missing here? Are the writers just ignoring this case since $(1)$ is not prime?

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The formulation allows them to ignore that case as they say that "the $p_i$ are precisely the prime ideals which occur in ..." –  Hagen von Eitzen Oct 2 '12 at 16:40
    
So I thought, but I wanted another opinion, so that I hadn't misunderstood. Thank you –  harajm Oct 2 '12 at 16:52
    
Off topic: I don't think the primary decomposition is that important to learn, especially for beginners. –  user18119 Oct 2 '12 at 23:35
    
Off topic: I think the primary decomposition is very important to learn, especially for beginners. –  user26857 Oct 10 '12 at 20:29

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