minimal ideals in a Noetherian ring

Question

I'm reading the "Advanced modern algebra" book (second edition), and I'm confused with minimal prime ideals.

By Theorem 6.116 (Lasker-Noether II) the associated prime ideals are uniquely determined by an ideal $I$.

This theorem does not care about isolated prime ideals and embedded ideals.

However, by Corollary 6.120(i), it says that only isolated prime ideals are uniquely determined by $I$.

This corollary implies that embedded associated prime ideals of $I$ are not unique but the Theorem says they are unique.

I'm confused!

Answer 1

Theorem 6.116 says that the associated primes of an irredundant primary decomposition are uniquely determined.

Corollary 6.120 says that the isolated primes of a normal primary decomposition are uniquely determined.

Notice these theorems make statements about different types of decompositions. Let's assume you have a normal primary decomposition (so that both statements apply.)

The first theorem says that there is a unique collection of primary ideals associated with $I$. The second statement says that there is a unique collection of isolated primes associated with $I$.

From this, you should not conclude "the embedded primes are not uniquely determined." To the contrary, you also know now that they are uniquely determined too.

These two statements together just say that

"There is a special class of primes associated to $I$, and they partition into subclasses we call isolated or embedded. It doesn't matter what primary decompositions you use to compute them because they all agree on which ones are isolated and which ones are embedded for $I$."