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By Theorem 4.10 of Atiyah's book which states that

"Let $a$ be a decomposable ideal, let $a=\cap_{i=1}^{n}q_i$ be a minimal primary decomposition of $a$, let $\{p_{i_1},...,p_{i_m}\}$ be an isolated set of prime ideals of $a$. Then $q_{i_1}\cap...\cap q_{i_m}$ is independent of the decomposition."

we know that intersection of isolated primary ideals are uniquely determined. Corollary 4.11 states that each of the isolated primary ideal is uniquely determined. I cannot conclude this corollary.

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A singleton set containing a minimal prime ideal belonging to $\mathfrak a$ is an isolated set.

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