{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]] and let $R[x]$ be the [[polynomial ring|0Q9EZS90GY]] in a central variable $x$ over $R$.\nLet $J(R[x])$ be the [[Jacobson radical|XA0DCNH7WY]] of $R[x]$ and let $N = J(R[x]) \\cap R$.\nThen $N$ is a [[nil ideal|NZ1G9WRWFB]] of $R$ and $J(R[x]) = N[x]$, the set of polynomials with all coefficients in $N$.\n\n## Remarks\n\nThe harder part is to show that $J(R[x])$ is generated by its constant polynomials.\nThe nil part is then direct: for $a \\in N$, the element $ax$ lies in $J(R[x])$, so $1 - ax$ is a [[unit|019NFGFPR5]] of $R[x]$.\nIts inverse must be the power series $\\sum_{n \\ge 0} a^n x^n$, which is a polynomial only if $a^n = 0$ for large $n$.\nExample: for a commutative ring $R$, $J(R[x])$ is $\\mathfrak{N}[x]$, where $\\mathfrak{N}$ is the [[nilradical|QQDN6EYQNJ]] of $R$; for instance $J(\\Z[x]) = 0$, although $\\Z$ has many maximal ideals.\nIn particular, if $R$ has no nonzero nil ideals, then $R[x]$ is [[semiprimitive|1B2DBKV1C9]].\n\n## Explanations\n\n### L1 human\n\nThe Jacobson radical of a ring of polynomials is always made of polynomials whose coefficients are nilpotent.\nSo adjoining a variable kills most of the radical: only the nilpotent part survives.\n\n### L2 human\n\nThe theorem is a standard tool for producing semiprimitive rings, since $R[x]$ is semiprimitive whenever $R$ has no nonzero nil ideals.\nIt also connects the Jacobson radical with nil ideals and with the [[Köthe conjecture|RT2D9JEVRR]], which in Krempa's form asks whether $J(R[x]) = R[x]$ for every nil ring $R$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "XA0DCNH7WY",
      "0SDC934R52"
    ],
    "kind": "statement",
    "msc": [
      "16N20",
      "16S36"
    ],
    "name": "Amitsur's theorem on the radical of polynomial rings",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jacobson_radical"
      }
    ],
    "tag": "VW2H7873YM",
    "uses": [
      "0894J980RK",
      "0Q9EZS90GY",
      "XA0DCNH7WY",
      "NZ1G9WRWFB",
      "019NFGFPR5"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/VW2H7873YM/",
  "links": [
    {
      "field": "about",
      "target": "XA0DCNH7WY"
    },
    {
      "field": "about",
      "target": "0SDC934R52"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    },
    {
      "field": "uses",
      "target": "0Q9EZS90GY"
    },
    {
      "field": "uses",
      "target": "XA0DCNH7WY"
    },
    {
      "field": "uses",
      "target": "NZ1G9WRWFB"
    },
    {
      "field": "uses",
      "target": "019NFGFPR5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VW2H7873YM.md",
  "msc": [
    "16N20",
    "16S36"
  ],
  "name": "Amitsur's theorem on the radical of polynomial rings",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VW2/VW2H7873YM.md",
  "status": null,
  "tag": "VW2H7873YM",
  "uri": "m2://VW2H7873YM",
  "url": "https://tamnd.github.io/math2/wiki/amitsurs-theorem-on-the-radical-of-polynomial-rings/",
  "writing": {
    "tag": "VW2H7873YM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}