While studying primary decomposition from Atiyah-Macdonald I came across this problem:
For any prime ideal $p$ in a ring A, let $S_p(0)$ denote the kernel of the homomorphism $A \rightarrow A_p$ Prove that
i) $S_p(0) S \subset p$
ii) $r(S_p(0)) = p \iff p$ is a minimal prime ideal of $A$.
While I was able to solve first one easily, I am having a hard time understanding the meaning of second problem. Specially the quote
$p$ is a minimal prime ideal of $A$.
Here I can't understand, the meaning of minimal prime ideal. What I have read so far says minimal prime ideal over an ideal $a$ of the ring $A$ is the set of minimal prime elements of minimal primary decomposition of the ideal $a$. But in the question no such ideal is given. I have read in some place that they are trying to imply here $p$ is a minimal prime ideal of $A$ means it's a minimal prime ideal over the ideal $(0)$. But in this case that will imply $(0)$ ideal is decomposable. But correct me if I am wrong, $(0)$ ideal is not always decomposable. So it will be great if you can help me understanding the question.
Thanks
Prime ideals form a poset. A minimal prime ideal is a minimal element in that poset.