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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $I$ be an [[ideal|0MYSA6MYNA]] of $R$, and let $P_1, \\ldots, P_n$ be finitely many [[prime ideals|06HC6HK0Z8]] of $R$, with $n \\ge 1$.\nIf $I \\subseteq P_1 \\cup \\cdots \\cup P_n$, then $I \\subseteq P_i$ for some $i$.\n\n## Remarks\n\nThe proof is by induction on $n$.\nIf $I$ is contained in no $P_i$, the induction hypothesis gives, for each $i$, an element $x_i \\in I$ that lies in no $P_j$ with $j \\ne i$; one may assume $x_i \\in P_i$.\nThen $y = x_1 + x_2 x_3 \\cdots x_n$ lies in $I$ but in none of the $P_i$, using that $P_1$ is prime for the term $x_2 \\cdots x_n$.\nThe same proof works when at most two of the $P_i$ are not prime.\nPrimality is needed: in $R = \\F_2[x, y]/(x, y)^2$, the maximal ideal $(x, y) = \\set{0, x, y, x + y}$ is the union of the three ideals $(x)$, $(y)$, $(x + y)$, but it is contained in none of them.\nFiniteness is needed: in $k[x, y]$ for a field $k$, every nonzero element of $(x, y)$ has an irreducible factor vanishing at the origin.\nSo $(x, y)$ is the union of infinitely many principal prime ideals, none of which contains it.\n\n## Explanations\n\n### L1 human\n\nAn ideal cannot hide inside a finite union of prime ideals without being inside one of them.\nSo if an ideal is not contained in any of finitely many primes, one can find a single element of the ideal that avoids all of them at once.\nThis is the form in which the lemma is used most often.\n\n### L2 human\n\nPrime avoidance is a tool for choosing good elements.\nFor example, let $M$ be a finitely generated module over a Noetherian ring, and let $I$ be an ideal whose elements are all zero divisors on $M$.\nSince [[zero divisors are the union of associated primes|S37DQGW3TZ]], $I$ lies in the union of the finitely many [[associated primes|ERFZ6S8X0X]] of $M$.\nSo $I$ lies in one associated prime.\nIn dimension theory the lemma lets one choose elements outside all minimal primes, which is how [[regular sequences|69R5R2FCS5]] and [[systems of parameters|95QTEPMDA4]] are built one element at a time.\nIf $R$ contains an infinite field, the conclusion holds even when none of the ideals $P_i$ is prime, by a vector space argument.\n",
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