Prime ideals and Zorn's lemma

Question

Let $A$ be a ring, $x$ a nonzero element of $A$ and consider the annihilator of $x$, i.e $Ann(x)$.

Now let $S$ denote the collection of all prime ideals of $A$ containing $Ann(x)$. It can be shown using Zorn's lemma that $S$ contains minimal elements (with respect the inclusion $\subseteq$).

My question is about the following argument:

Let $P$ be a minimal prime ideal in the set of all prime ideals that contain $Ann(x)$ and suppose $z \in P$. My question is: why does it follows that $z \in \sqrt{Ann(x)}$ ?

We know that $\sqrt{Ann(x)}$ consists of all prime ideals of $A$ that contain $Ann(x)$.

So let $P$ be a minimal prime ideal in the set of all prime ideals that contain $Ann(x)$ and assume $z \in P$.

Now let $J$ be any prime ideal of $A$ that contains $Ann(x)$. We need to show that $z \in J$.

Since $P$ is minimal with respect the inclusion we can't have that $J$ is properly contained in $P$. Why does it follows then that $P$ must contain $J$ ?

Can you please explain?

Answer 1

Unless I'm doing something silly, I think the claim is incorrect.

Let $A=\mathbb{Z}/36\mathbb{Z}$, and let $x=6+36\mathbb{Z}$.

The annihilator of $x$ consists of all $m+36\mathbb{Z}$ such that $6m$ is divisible by $36$, equivalently, such that $m$ is divisible by $6$; that is, $\mathrm{Ann}(6+36\mathbb{Z}) = (x)$.

The prime ideals of $A$ are the images of the prime ideals of $\mathbb{Z}$ that contain $36$, namely $(2+36\mathbb{Z})$ and $(3+36\mathbb{Z})$; they are both minimal elements of the set of all prime ideals that contain $(x)$. Now take $z=2+36\mathbb{Z}$. Then $z$ is not in the radical of the annihilator: if $2^n+36\mathbb{Z} \in 6+36\mathbb{Z}$, then $9|2^n-6$ and $4|2^n-6$. The latter requires $n=1$, in which case the former cannot hold.

Answer 2

It doesn't follow and, in fact, your statement is not true. Consider the ring $R=\mathbb{Z}/6\mathbb{Z}$, which has minimal prime ideals $P=(2)$ and $Q=(3)$. Then $P,Q$ are both minimal primes containing ${\rm Ann}(1)=\{0\}$. However, none of $2,4\in P$ or $3\in Q$ are in $\sqrt{{\rm Ann}(1)}=\{0\}$.