The set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal 
I'm asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal. 

I can't even start in the proof, I've just defined my set but cant move on construction the ideal !
 A: Let $Z$ be the set of all zero divisors in a commutative ring $R$. Note that $R \setminus Z$ is closed under multiplication.
Indeed, take $a, b \in R \setminus Z$ and assume $ab \in Z$. Then there exists nonzero $c \in R$ such that $$0 = (ab)c = a(bc)$$
If $bc \ne 0$ then $a(bc) = 0$ implies that $a$ is a zero divisor. On the other hand, if $bc = 0$ then $b$ is a zero divisor. A contradiction.
Now consider the family of all ideals in $R$ contained in $Z$:$$\mathcal{S} = \{I \trianglelefteq R : I \subseteq Z\}$$
Order $\mathcal{S}$ by inclusion. We have $\{0\} \in \mathcal{S}$, and for every chain $\mathcal{L}$ in $\mathcal{S}$ we have that $\bigcup \mathcal{L}$ is an upper bound for $\mathcal{L}$ which is contained in $\mathcal{S}$. Zorn's lemma implies that $\mathcal{S}$ has a maximal element $P$. We claim that $P$ is prime.
Let $A, B$ be two ideals in $R$ such that $AB \subseteq P$. Assume $A, B \not\subseteq P$. Then $P + A$ and $P + B$ are two ideals in $R$ which strictly contain $P$. Therefore, $P + A$ and $P + B$ intersect $R \setminus Z$. Let $p_1 + a \in (P + A) \cap R \setminus Z$ and $p_2 + b \in (P + B) \cap R \setminus Z$.
Since $R \setminus Z$ is closed under multiplication we have:
$$R \setminus Z \ni (p_1 + a)(p_2 + b) = p_1p_2 + p_1b + ap_2 + ab \in P$$
which is a contradiction since $P \subseteq Z$.
Therefore $P$ is a prime ideal of $R$ contained in $Z$.
A: One way to approach this is to combine these two lemmas:


*

*In a commutative ring, there exist minimal prime ideals.

*In a commutative ring with identity, minimal primes consist entirely of zero divisors (I use the convention that $0$ is a zero divisor here.) 
