Abstract Algebra – Zero Divisors in a Commutative Ring and Prime Ideals

Question

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 !

Answer 1

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 $Z$.