Prime avoidance lemma

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.

Statement and proof

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 $I_1,I_2,\dots ,I_n,n\ge 1$ be ideals such that $I_i$ are prime ideals for $i\ge 3$. If E is not contained in any of $I_i$'s, then E is not contained in the union $\cup I_i$. Proof by induction on n: The idea is to find an element that is in E and not in any of $I_i$'s. The basic case n = 1 is trivial. Next suppose n ≥ 2. For each i, choose

$z_i\in E-{\cup }_{j\ne i}{I}_j$

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

$z=z_1\dots z_{n-1}+z_n$.

Then z is in E but not in any of $I_i$'s. Indeed, if z is in $I_i$ for some $i\le n-1$, then $z_n$ is in $I_i$, a contradiction. Suppose z is in $I_n$. Then $z_1\dots z_{n-1}$ is in $I_n$. If n is 2, we are done. If n > 2, then, since $I_n$ is a prime ideal, some $z_i,i<n$ is in $I_n$, a contradiction.

E. 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 $I\subset ̸{\mathfrak{𝔭}}_i$ for each i, then there exists some y in J such that $x+y\in ̸{\mathfrak{𝔭}}_i$ 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 $x\in ̸{\mathfrak{𝔭}}_i$ 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 $x+y\in I-\underset{1}{\overset{r-1}{\cup }}{\mathfrak{𝔭}}_i$. If $x+y$ is not in ${\mathfrak{𝔭}}_r$, we are done. Otherwise, note that $J\subset ̸{\mathfrak{𝔭}}_r$ (since $x\in {\mathfrak{𝔭}}_r$) and since ${\mathfrak{𝔭}}_r$ is a prime ideal, we have:

${\mathfrak{𝔭}}_r\supset ̸J{\mathfrak{𝔭}}_1\cdots {\mathfrak{𝔭}}_{r-1}$.

Hence, we can choose $y^{\prime }$ 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 $x+y+y^{\prime }$ has the required property. $◻$

Application

Let A be a Noetherian ring, I an ideal generated by n elements and M a finite A-module such that $IM\ne M$. Also, let $d={\mathrm{depth}}_A(I,M)$ = the maximal length of M-regular sequences in I = the length of every maximal M-regular sequence in I. Then $d\le n$; 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 $I\subset ̸{\mathfrak{𝔭}}_i$ 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 $x_1\in ̸\underset{1}{\overset{r}{\cup }}{\mathfrak{𝔭}}_i$ = 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, ${\mathrm{depth}}_{A/(x_1)}(I/(x_1),M/x_{1}M)\le n-1$. The claim now follows.

References

• 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.