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Prime avoidance lemma

Result concerning ideals of commutative rings

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Source updated March 21, 2026

In algebra, the prime avoidance lemma says that if an ideal I in a commutative ring R is contained in a union of finitely many prime ideals P_i's, then it is contained in P_i for some i.

There are many variations of the lemma (cf. Hochster); for example, if the ring R contains an infinite field or a finite field of sufficiently large cardinality, then the statement follows from a fact in linear algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces.

01Statement and proof

The following statement and argument are perhaps the most standard.

Theorem (Prime Avoidance Lemma): Let E be a subset of commutative ring R that is an additive subgroup of R and is multiplicatively closed. (In particular, E could be a subring or ideal of R.) Let $I_{1},I_{2},\dots ,I_{n},n\geq 1$ be ideals such that $I_{i}$ are prime ideals for $i\geq 3$ . If E is not contained in any of the $I_{i}$ , then E is not contained in the union ${\textstyle \bigcup I_{i}$ .

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

$z_{i}\in E\setminus \bigcup _{j\neq i}I_{j}$ ,

where each of the sets on the right is nonempty by the inductive hypothesis. We can assume $z_{i}\in I_{i}$ for all i; otherwise, there is some $z_{k}$ among them that avoids all of the $I_{i}$ , and we are done. Put

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

Because E is closed under addition and multiplication, z is in E by construction. We claim that z is not in any of the $I_{i}$ . Indeed, if $z\in I_{i}$ for some $i\leq n-1$ , then $z_{n}\in I_{i}$ , a contradiction. Next suppose $z\in I_{n}$ . Then $z_{1}\cdots z_{n-1}\in I_{n}$ . If $n=2$ , this is already a contradiction. If $n>2$ , then, since $I_{n}$ is a prime ideal, $z_{i}\in I_{n}$ for some $i\leq n-1$ , again a contradiction. $\square$

02E. Davis' prime avoidance

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

Theorem, Let A be a ring, ${\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}$ prime ideals, x an element of A and J an ideal. For the ideal $I=xA+J$ , if $I\not \subset {\mathfrak {p}}_{i}$ for each i, then there exists some y in J such that $x+y\not \in {\mathfrak {p}}_{i}$ for each i.

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

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

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

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

Application

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

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

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

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Sources and credits

This article is adapted from the Wikipedia article “Prime avoidance lemma”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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