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  "body": "\n## Statement\n\nLet $L$ be a [[distributive lattice|0HM0HQCK6N]].\nLet $I$ be an [[ideal|5GW2T2AM1A]] of $L$ and let $F$ be a [[filter|Z15FXM9WM1]] of $L$ with $I \\cap F = \\emptyset$.\nThen there is a [[prime ideal|QKWVXJWHP6]] $P$ of $L$ such that $I \\subseteq P$ and $P \\cap F = \\emptyset$.\n\n## Remarks\n\nThe usual proof applies [[Zorn's lemma|04RXTQA5ZA]] to the set of ideals that contain $I$ and are disjoint from $F$, ordered by inclusion.\nLet $P$ be a maximal one, and suppose $x \\wedge y \\in P$ with $x \\notin P$ and $y \\notin P$.\nBy maximality, the ideals generated by $P \\cup \\set{x}$ and by $P \\cup \\set{y}$ meet $F$, so $f \\le p \\vee x$ and $g \\le q \\vee y$ for some $f, g \\in F$ and $p, q \\in P$.\nThen $f \\wedge g \\le (p \\vee q) \\vee (x \\wedge y)$ by distributivity, so $f \\wedge g \\in P \\cap F$, a contradiction.\n\nExample: in the lattice of divisors of $12$, with $\\gcd$ as meet and $\\lcm$ as join, take $I = \\set{1, 2}$ and $F = \\set{3, 6, 12}$.\nThe set $P = \\set{1, 2, 4}$ of divisors not divisible by $3$ is a prime ideal with $I \\subseteq P$ and $P \\cap F = \\emptyset$.\n\nDistributivity is necessary.\nIn the [[diamond lattice M3|JQ6FJH2TP8]] with middle elements $a, b, c$, the proper ideals are $\\set{0}$ and $\\set{0, x}$ for a middle element $x$.\nNone of them is prime: for instance $a \\wedge b = 0$, while neither $a$ nor $b$ lies in $\\set{0, c}$.\nSo the ideal $\\set{0}$ and the filter $\\set{1}$ admit no separating prime ideal.\n\nThe [[Boolean prime ideal theorem|STPBXBKSZP]] is the special case of Boolean algebras.\nOver ZF set theory, the two statements are equivalent, and both are strictly weaker than the [[axiom of choice|0HTGB13FE2]].\n\n## Explanations\n\n### L1 human\n\nAn ideal is a family of \"small\" elements and a filter is a family of \"large\" elements.\nIf they do not overlap, the theorem finds a prime ideal that keeps all the small ones and avoids all the large ones.\nPrime ideals are the \"points\" of a distributive lattice, so the theorem says there are enough points to separate things.\n\n### L2 human\n\nThe theorem is the key step in representing a distributive lattice as a lattice of sets: map $a$ to the set of prime filters that contain it.\nSeparating $a \\not\\le b$ by a prime ideal containing $b$ and missing $a$ shows that this map is injective.\nThis is the basis of Stone's representation theorem for distributive lattices and of [[Priestley duality|SA22SEC75K]].\nIt needs a form of choice, and over ZF it is equivalent to the [[Boolean prime ideal theorem|STPBXBKSZP]].\n",
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