ac.commutative algebra - Atiyah-MacDonald, exercise 7.19 - "decomposition using irreducible ideals"

An ideal $mathfrak{a}$ is called irreducible if $mathfrak{a} = mathfrak{b} cap mathfrak{c}$ implies $mathfrak{a} = mathfrak{b}$ or $mathfrak{a} = mathfrak{c}$. Atiyah-MacDonald Lemma 7.11 says that in a Noetherian ring, every ideal is a finite intersection of irreducible ideals. Exercise 7.19 is about the uniqueness of such a decomposition.

7.19. Let $mathfrak{a}$ be an ideal in a noetherian ring. Let
$$mathfrak{a} = cap_{i=1}^r mathfrak{b}_i = cap_{j=1}^s mathfrak{c}_j$$ be two minimal decompositions of $mathfrak{a}$ as an intersection of irreducible ideals. [I assume minimal means that none of the ideals can be omitted from the intersection.] Prove that $r = s$ and that (possibly after reindexing) $sqrt{mathfrak{b}_i} = sqrt{mathfrak{c}_i}$ for all $i$.

Comments: It's true that every irreducible ideal in a Noetherian ring is primary (Lemma 7.12), but I don't think our result follows from the analogous statement about primary decomposition. For example, here is Example 8.6 from Hassett's $textit{Introduction to Algebraic Geometry}$.

8.6 Consider $I = (x^2, xy, y^2) subset k[x,y]$. We have $$I = (y, x^2) cap (y^2, x) = (y+x, x^2) cap (x, (y+x)^2),$$ and all these ideals (other than $I$) are irreducible.

If my interpretation of "minimal" is correct, then this is a minimal decomposition using irreducible ideals, but it is not a minimal primary decomposition, because the radicals are not distinct: they all equal $(x,y)$.

There is a hint in the textbook: Show that for each $i = 1, ldots, r$, there exists $j$ such that $$mathfrak{a} = mathfrak{b}_1 cap cdots cap mathfrak{b}_{i-1} cap mathfrak{c}_j cap mathfrak{b}_{i+1} cap cdots cap mathfrak{b}_r.$$ I was not able to prove the hint.

I promise this exercise is not from my homework.

Update. There doesn't seem to be much interest in my exercise. I've looked at various solution sets on the internet, and I believe they all make the mistake of assuming that a minimal irreducible decomposition is a minimal primary decomposition. Does anyone know of a reference which discusses irreducible ideals? Some google searches have produced Hassett's book that I mention above and not much else.

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