I have a question regarding 'a converse of the principal ideal theorem' in A course in commutative algebra by George Kemper (theorem 7.8). It says:
Let $R$ be a Noetherian ring, and $P \in $ Spec $(R)$ a prime ideal of height $n$. Then there exist $a_1, . . . , a_n \in R$ such that $P$ is minimal over $I= (a_1, . . . , a_n)$.
As far as I understand, '$P$ is minimal over $I$' means that $P$ is the minimal ideal in the set of ideals that contain $I$. But does this not mean the same as $P=I$? I guess not, since it is written in this way, but I cannot see how $P$ could be the minimal without being equal to $I$, because clearly also $I$ contains $I$ so this would always be smaller or equal than $P$.