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 Pi's, then it is contained in Pi 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
Proof by induction on n: The idea is to find an element that is in E and not in any of
where the set on the right is nonempty by inductive hypothesis. We can assume
Then z is in E but not in any of