Prime avoidance lemma

Statement

Let $R$ be a commutative unital ring. Let $I_1,I_2,\dots ,I_n$ and $J$ be ideals of $R$, such that $J\subset {\bigcup }_j{I}_j$. Then, if $R$ contains an infinite field or if at most two of the $I_j$s are not prime, then $J$ is contained in one of the $I_j$s.

Graded version

If $R$ is graded, and $J$ is generated by homogeneous elements of positive degree, then it suffices to assume that the homogeneous elements of $J$ are contained in ${\bigcup }_j{I}_j$.

Importance

The prime avoidance lemma is useful for establishing dichotomies; in particular, if $J$ is an ideal which is not cintained in any of the $I_j$s, then $J$ has an element which is contained in none of the $I_j$s.