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.
Prime avoidance can be used to show the following fundamental result on regular sequences:
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 $M$.
It can also be used to show the following:
Regular local rings are integral.
The proof of the First and Second Uniqueness Theorems of Primary Decomposition uses prime avoidance lemma in an essential way.
See Atiyah-Mac Donald, Introduction to Commutative Algebra, Chapter 4 (in that book prime avoidance lemma is referred as Proposition 1.11).
Theorem. For a given commutative ring $R$, then $Min(R)$, the set of minimal prime ideals of $R$, is a finite set if and only if no minimal prime ideal of $R$ is contained in the union of the remaining minimal primes.
Sketch of Proof. The implication $\Rightarrow$ of the above nice result is deduced from the prime avoidance lemma. The reverse implication is deduced from the fact that $Min(R)$ is quasi-compact with respect to the flat topology.
Remember that the collection of $V(f)=\{\mathfrak{p}\in Spec(R):f\in\mathfrak{p}\}$ with $f\in R$ forms a sub-base for the opens of the flat topology over $Spec(R)$.
For more details on the flat topology see arXiv:1609.00947 or arXiv:1503.04299.
I just stumbled upon the 4-year-old question. Let me add another application to the list.
Claim. Let $A$ be a commutative Noetherian ring. If primes $\mathfrak p, \mathfrak r$ satisfy $\mathfrak p < \mathfrak r$, then the interval $(\mathfrak p, \mathfrak r)$ contains infinitely many primes or zero primes.
Chat. Here I am working in the ordered set of primes, so $\mathfrak p<\mathfrak r$ means $\mathfrak p\subsetneq \mathfrak r$. Here $(\mathfrak p, \mathfrak r)$ denotes the set of primes strictly intermediate to $\mathfrak p$ and $\mathfrak r$. Finally, this question about the structure of intervals in the ordered set of primes is asked in order to understand Spec($A$).
Reasoning for Claim. Suppose otherwise that $(\mathfrak p, \mathfrak r)$ is finite and nonempty. Shrink this interval to a minimal such one, and we end up with an interval containing some $\mathfrak q\neq \mathfrak p, \mathfrak r$ such that $\mathfrak p\prec \mathfrak q\prec \mathfrak r$ and $(\mathfrak p,\mathfrak r)$ is finite. (The notation $\mathfrak p\prec \mathfrak q$ indicates covering, which means that $\mathfrak p$ is included in $\mathfrak p$ and $(\mathfrak p,\mathfrak q)$ is empty.) When we have shrunk to a minimal such interval, no two distinct primes strictly between $\mathfrak p$ and $\mathfrak r$ will be comparable.
Factor by $\mathfrak p$ to reduce to the case $\mathfrak p=(0)$. Then localize at $\mathfrak r$ to reduce to the case where $\mathfrak r$ is maximal. Now we have $(0)\prec \mathfrak q\prec \mathfrak r$ and that $\mathfrak r$ is maximal. In this reduced case we are trying to show that it is impossible for a Noetherian integral domain of Krull dimension $2$ to have only finitely many primes. This is where Prime Avoidance comes in. Assume that $\mathfrak q_1, \ldots, \mathfrak q_n$ is a complete list of the primes strictly between $(0)$ and $\mathfrak r$ (i.e., height-$1$ primes). By Prime Avoidance, it is impossible to have $\mathfrak r$ contained in $\cup_{i=1}^n \mathfrak q_i$, so choose $a\in \mathfrak r - \cup_{i=1}^n \mathfrak q_i$. By the Krull Principal Ideal Theorem, any prime minimal over $(a)$ must have height $1$, so must be one of the $\mathfrak q_i$'s. But we specifically chose $a$ so that it belongs to none of them, so we are done. $\Box$
The Noetherianness hypothesis is used to allow us to invoke the Krull Principal Ideal Theorem.
