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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $\\operatorname{Prim}(R)$ be the set of its [[primitive ideals|M0JDQR0V5S]].\nFor a subset $S \\subseteq \\operatorname{Prim}(R)$, its *kernel* is $k(S) = \\bigcap_{P \\in S} P$, with $k(\\varnothing) = R$.\nFor a [[two-sided ideal|0DDA6CY7GJ]] $I$ of $R$, its *hull* is $h(I) = \\set{P \\in \\operatorname{Prim}(R) : P \\supseteq I}$.\nThe *Jacobson topology*, or *hull-kernel topology*, on $\\operatorname{Prim}(R)$ is the [[topology|0CNQXVBHCX]] whose closure operator is\n$$\\overline{S} = h(k(S)).$$\nEquivalently, its closed sets are exactly the hulls $h(I)$ of the two-sided ideals $I$ of $R$.\n\n## Remarks\n\nThe map $S \\mapsto h(k(S))$ satisfies the Kuratowski closure axioms, so it does define a topology.\nThe space $\\operatorname{Prim}(R)$ is $T_0$: two distinct primitive ideals $P \\ne Q$ have different closures $h(P) \\ne h(Q)$.\nExample: for $R = \\Z$ the primitive ideals are the ideals $p\\Z$ with $p$ prime.\nThe kernel of an infinite set of them is $0$, so its closure is all of $\\operatorname{Prim}(\\Z)$; the closed sets are the finite sets and the whole space.\nNon-example of a closed set: $\\set{p\\Z : p \\text{ an odd prime}}$ is not closed, because its closure contains $2\\Z$.\nFor a commutative ring $R$, the primitive ideals are the maximal ideals and the Jacobson topology is the subspace topology from the Zariski topology on the prime spectrum.\n",
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