{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nThe *spectrum* of $R$ is the set\n$$\\operatorname{Spec} R = \\set{\\mathfrak p : \\mathfrak p \\text{ is a prime ideal of } R},$$\nequipped with the [[Zariski topology|8SQK7RTB63]], whose closed sets are the sets $V(E) = \\set{\\mathfrak p \\in \\operatorname{Spec} R : E \\subseteq \\mathfrak p}$ for subsets $E \\subseteq R$.\n\n## Remarks\n\nA [[ring homomorphism|0VTH8KPCG2]] $f \\colon R \\to A$ induces the map $\\operatorname{Spec} A \\to \\operatorname{Spec} R$, $\\mathfrak q \\mapsto f^{-1}(\\mathfrak q)$.\nIt is continuous, since the preimage of $V(E)$ is $V(f(E))$.\n\nExample: $\\operatorname{Spec} \\Z$ consists of $(0)$ and the ideals $p\\Z$ for the prime numbers $p$.\nThe spectrum of a field $k$ is a single point $(0)$.\nThe spectrum of $\\Z/n\\Z$ for $n \\ge 2$ is the finite set of ideals $p\\Z/n\\Z$ for the primes $p$ dividing $n$.\n\nNon-example: the ring $R$ itself is not a point of $\\operatorname{Spec} R$, because a [[prime ideal|00MHHZ9SNE]] is by definition a proper ideal.\nSo the zero ring has empty spectrum.\n\nEvery nonzero ring has a [[maximal ideal|0PV74TR5FP]], which is prime, so $\\operatorname{Spec} R \\ne \\emptyset$ when $R \\ne 0$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FB45BHHA2J",
    "kind": "definition",
    "msc": [
      "14A05",
      "13A15"
    ],
    "name": "Spectrum of a ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectrum_of_a_ring"
      }
    ],
    "tag": "QJAXM2SF1Z",
    "uses": [
      "0310REJJY8",
      "00MHHZ9SNE",
      "8SQK7RTB63",
      "0VTH8KPCG2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/QJAXM2SF1Z/",
  "links": [
    {
      "field": "defines",
      "target": "FB45BHHA2J"
    },
    {
      "field": "uses",
      "target": "0310REJJY8"
    },
    {
      "field": "uses",
      "target": "00MHHZ9SNE"
    },
    {
      "field": "uses",
      "target": "8SQK7RTB63"
    },
    {
      "field": "uses",
      "target": "0VTH8KPCG2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QJAXM2SF1Z.md",
  "msc": [
    "14A05",
    "13A15"
  ],
  "name": "Spectrum of a ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QJA/QJAXM2SF1Z.md",
  "status": null,
  "tag": "QJAXM2SF1Z",
  "uri": "m2://QJAXM2SF1Z",
  "url": "https://math2.tamnd.com/wiki/spectrum-of-a-ring-definition/",
  "writing": {
    "tag": "QJAXM2SF1Z",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}