I was wondering if the prime avoidance lemma is very useful or just a nice result. So far I know just only one application: let $R$ be a commutative noetherian ring and $I$ be a proper ideal of $R$. If $I$ consists only of zero divisors of $R$, then $I$ is contained in some associated prime ideal of $(0)$.


So my question is: are there other applications of the prime avoidance lemma in commutative algebra? Thanks in advance for your answers.



Applications of the prime avoidance lemma
Prime avoidance can be used to show the following fundamental result on regular sequences:



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> If $R$ is a noetherian ring, $\mathfrak{a}\subseteq R$ is an ideal, and $M$ is an $R$-module of finite type, then every maximal $M$-sequence in $\mathfrak{a}$ has length equal to the $\mathfrak{a}$-depth of $M$.
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It can also be used to show the following:



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> Regular local rings are integral.
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