Lemma 10.27.12. Let $R$ be a ring.

An ideal maximal among the ideals which do not contain a nonzerodivisor is prime.

If $R$ is nonzero and every nonzero prime ideal in $R$ contains a nonzerodivisor, then $R$ is a domain.

Lemma 10.27.12. Let $R$ be a ring.

An ideal maximal among the ideals which do not contain a nonzerodivisor is prime.

If $R$ is nonzero and every nonzero prime ideal in $R$ contains a nonzerodivisor, then $R$ is a domain.

**Proof.**
Consider the set $S$ of nonzerodivisors. It is a multiplicative subset of $R$. Hence any ideal maximal with respect to not intersecting $S$ is prime, see Lemma 10.27.9. Thus, if every nonzero prime ideal contains a nonzerodivisor, then $(0)$ is prime, i.e., $R$ is a domain.
$\square$

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