Clarification on Notes about Minimal Prime Ideals over an Ideal $I$ of a Noetherian ring $R$
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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), $sqrt{I} = P_1 cap dotscap P_m$ for some primes $P_i$ such that $P_j notsubset P_i ; forall i neq j$ . 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 prime
