Prime avoidance lemma - Alchetron, The Free Social Encyclopedia

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 , … , I n , n ≥ 1 be ideals such that I i are prime ideals for i ≥ 3 . If E is not contained in any of I i 's, then E is not contained in the union ∪ 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 ∈ E − ∪ j ≠ i I j

where the set on the right is nonempty by inductive hypothesis. We can assume z i ∈ I i for all i; otherwise, some z i avoids all the I i 's and we are done. Put

z = z 1 … 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 ≤ n − 1 , then z n is in I i , a contradiction. Suppose z is in I n . Then z 1 … 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.

References

Prime avoidance lemma Wikipedia

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