The following statement and argument are perhaps the most standard.

Statement: Let E be a subset of R that is an additive subgroup of R and is multiplicatively closed. Let ${\displaystyle I_{1},I_{2},\dots ,I_{n},n\geq 1}$  be ideals such that ${\displaystyle I_{i}}$  are prime ideals for ${\displaystyle i\geq 3}$ . If E is not contained in any of ${\displaystyle I_{i}}$ 's, then E is not contained in the union ${\displaystyle \cup I_{i}}$ .

Proof by induction on n: The idea is to find an element that is in E and not in any of ${\displaystyle I_{i}}$ 's. The basic case n = 1 is trivial. Next suppose n ≥ 2. For each i, choose

${\displaystyle z_{i}\in E-\cup _{j\neq i}I_{j}}$ 

where the set on the right is nonempty by inductive hypothesis. We can assume ${\displaystyle z_{i}\in I_{i}}$  for all i; otherwise, some ${\displaystyle z_{i}}$  avoids all the ${\displaystyle I_{i}}$ 's and we are done. Put

${\displaystyle z=z_{1}\dots z_{n-1}+z_{n}}$ .

Then z is in E but not in any of ${\displaystyle I_{i}}$ 's. Indeed, if z is in ${\displaystyle I_{i}}$  for some ${\displaystyle i\leq n-1}$ , then ${\displaystyle z_{n}}$  is in ${\displaystyle I_{i}}$ , a contradiction. Suppose z is in ${\displaystyle I_{n}}$ . Then ${\displaystyle z_{1}\dots z_{n-1}}$  is in ${\displaystyle I_{n}}$ . If n is 2, we are done. If n > 2, then, since ${\displaystyle I_{n}}$  is a prime ideal, some ${\displaystyle z_{i},i<n}$  is in ${\displaystyle I_{n}}$ , a contradiction.

There is the following variant of prime avoidance due to E. Davis.

Proof:^[3] We argue by induction on r. Without loss of generality, we can assume there is no inclusion relation between the ${\displaystyle {\mathfrak {p}}_{i}}$ 's; since otherwise we can use the inductive hypothesis.

Also, if ${\displaystyle x\not \in {\mathfrak {p}}_{i}}$  for each i, then we are done; thus, without loss of generality, we can assume ${\displaystyle x\in {\mathfrak {p}}_{r}}$ . By inductive hypothesis, we find a y in J such that ${\displaystyle x+y\in I-\cup _{1}^{r-1}{\mathfrak {p}}_{i}}$ . If ${\displaystyle x+y}$  is not in ${\displaystyle {\mathfrak {p}}_{r}}$ , we are done. Otherwise, note that ${\displaystyle J\not \subset {\mathfrak {p}}_{r}}$  (since ${\displaystyle x\in {\mathfrak {p}}_{r}}$ ) and since ${\displaystyle {\mathfrak {p}}_{r}}$  is a prime ideal, we have:

${\displaystyle {\mathfrak {p}}_{r}\not \supset J\,{\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{r-1}}$ .

Hence, we can choose ${\displaystyle y'}$  in ${\displaystyle J\,{\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{r-1}}$  that is not in ${\displaystyle {\mathfrak {p}}_{r}}$ . Then, since ${\displaystyle x+y\in {\mathfrak {p}}_{r}}$ , the element ${\displaystyle x+y+y'}$  has the required property. ${\displaystyle \square }$ 

Application edit

Let A be a Noetherian ring, I an ideal generated by n elements and M a finite A-module such that ${\displaystyle IM\neq M}$ . Also, let ${\displaystyle d=\operatorname {depth} _{A}(I,M)}$  = the maximal length of M-regular sequences in I = the length of every maximal M-regular sequence in I. Then ${\displaystyle d\leq n}$ ; this estimate can be shown using the above prime avoidance as follows. We argue by induction on n. Let ${\displaystyle \{{\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}\}}$  be the set of associated primes of M. If ${\displaystyle d>0}$ , then ${\displaystyle I\not \subset {\mathfrak {p}}_{i}}$  for each i. If ${\displaystyle I=(y_{1},\dots ,y_{n})}$ , then, by prime avoidance, we can choose

${\displaystyle x_{1}=y_{1}+\sum _{i=2}^{n}a_{i}y_{i}}$ 

for some ${\displaystyle a_{i}}$  in ${\displaystyle A}$  such that ${\displaystyle x_{1}\not \in \cup _{1}^{r}{\mathfrak {p}}_{i}}$  = the set of zero divisors on M. Now, ${\displaystyle I/(x_{1})}$  is an ideal of ${\displaystyle A/(x_{1})}$  generated by ${\displaystyle n-1}$  elements and so, by inductive hypothesis, ${\displaystyle \operatorname {depth} _{A/(x_{1})}(I/(x_{1}),M/x_{1}M)\leq n-1}$ . The claim now follows.

• Mel Hochster, Dimension theory and systems of parameters, a supplementary note
• Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001.