# Clarification on Notes about Minimal Prime Ideals over an Ideal $I$ of a Noetherian ring $R$

Recently in my Algebra course, we defined the minimal prime ideals over an ideal $$I$$ of a Noetherian ring $$R$$, and then proved a result about them saying that $$\sqrt{I}$$ is the intersection of minimal primes over $$I$$. However, this is strange given the way he defines the minimal prime ideals over $$I$$. Below is the section of the notes, hopefully illustrating why it is confusing for me:

Now suppose $$I$$ is any ideal of a Noetherian ring R. By (2.13), $$\sqrt{I} = P_1 \cap \dots\cap P_m$$ for some primes $$P_i$$ such that $$P_j \not\subset P_i \; \forall i \neq j$$.

Note that if $$P$$ is prime containing $$I$$, then:

$$\prod_{i}P_i \leq P_1 \cap \dots \cap P_m = \sqrt{I} \leq P$$, and so: $$P_i \leq P$$

Definition (2.14): The minimal primes over an ideal $$I$$ of a Noetherian ring $$R$$ are these primes.

Lemma (2.15): Let $$I$$ be an ideal of a Noetherian ring $$R$$. Then $$\sqrt{I}$$ is the intersection of the minimal primers over $$I$$, and $$I$$ contains a finite product of them, possibly with repetitions.

This is what I have in my notes. The use of "these primes" in his definition made me think that he was defining the minimal primes to be the $$P_1, \dots, P_m$$ he mentioned earlier when talking about $$\sqrt{I}$$. However, the lemma then seems almost entirely redundant if this is how we define the minimal prime ideals.

Am I supposed to think that minimal primes over $$I$$ are just prime ideals of $$R$$ containing $$I$$ that are minimal with respect to inclusion, AND it can be shown that they have this special property relating to $$\sqrt{I}$$?

I'm just really confused about what this part of my notes means and what the point of this lemma is.

In case it's important, (2.13) says that for a Noetherian ring $$R$$, a radical ideal is the intersection of finitely many prime ideals.

Also, $$\sqrt{I} = \{{r \in R \mid r^n \in I \text{ for some } n\}}$$ is referred to as a radical ideal. We also defined the Nilradical and Jacobson Radical, but I don't know if these are included when we use the word "radical" here.

• Yes, (2.14) should have defined minimal primes over $I$ as these primes $P$ containing $I$ that is minimal with respect to inclusion. Commented Oct 22, 2018 at 15:17

Am I supposed to think that minimal primes over $$I$$ are just prime ideals of $$R$$ containing $$I$$ that are minimal with respect to inclusion, AND it can be shown that they have this special property relating to $$\sqrt{I}$$?
Yes. I think the point of 2.15 is that we know the radical is the intersection of all prime ideals over $$I$$, but in fact taking the minimal ones are enough. This follows trivially from your previous discussion.
In fact, you should think of the expression of $$\sqrt{I}=P_1\cap\cdots\cap P_n$$ as a special case of primary decomposition, where for radicals, it is actually prime decompositions. Then (treating it as primary decomposition) each $$P_i$$ is trivially $$P_i$$-primary, and we assumed that those $$P_i$$ are distinct. Therefore everyone is minimal, and there are no embedded primes.