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. ^[1]

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 ${\displaystyle I_{1},I_{2},\dots ,I_{n},n\geq 1}$ 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.

E. Davis' prime avoidance

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

Theorem —  ^[2] Let A be a ring, ${\displaystyle {\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}}$ prime ideals, x an element of A and J an ideal. For the ideal ${\displaystyle I=xA+J}$, if ${\displaystyle I\not \subset {\mathfrak {p}}_{i}}$ for each i, then there exists some y in J such that ${\displaystyle x+y\not \in {\mathfrak {p}}_{i}}$ for each i.

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

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.

Notes

1. Proof of the fact: suppose the vector space is a finite union of proper subspaces. Consider a finite product of linear functionals, each of which vanishes on a proper subspace that appears in the union; then it is a nonzero polynomial vanishing identically, a contradiction.
2. Matsumura 1986, Exercise 16.8.
3. Adapted from the solution to Matsumura 1986 , Exercise 1.6.

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.