{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R$ and $S$ be [[commutative rings|0GYGHKX834]], let $\\varphi \\colon R \\to S$ be a [[ring homomorphism|0GGE0Z0JQ1]] and let $s \\in S$.\nThen there is exactly one ring homomorphism $\\Phi \\colon R[x] \\to S$ such that $\\Phi(r) = \\varphi(r)$ for all $r \\in R$ and $\\Phi(x) = s$.\nIt is given by $\\Phi(a_0 + a_1 x + \\dots + a_n x^n) = \\varphi(a_0) + \\varphi(a_1) s + \\dots + \\varphi(a_n) s^n$.\n\n## Remarks\n\nWith $S = R$, $\\varphi$ the identity and $s = c$, the map $\\Phi$ is the [[evaluation homomorphism|0125QVNCFQ]] at $c$.\nIn the language of algebras: for each commutative $R$-algebra $A$ and each $a \\in A$, there is exactly one $R$-algebra homomorphism $R[x] \\to A$ that sends $x$ to $a$.\nThis property characterizes $R[x]$ up to a unique isomorphism.\n\n## Explanations\n\n### L1 human\n\nA ring homomorphism from $R[x]$ is fixed by two choices: what it does on the coefficients, and where it sends $x$.\nEvery pair of choices works, and it gives exactly one homomorphism.\nEvaluation at a number is the main example.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "13F20"
    ],
    "name": "Universal property of the polynomial ring",
    "role": "theorem",
    "sources": [
      {
        "locator": "Categorical characterization",
        "url": "https://en.wikipedia.org/wiki/Polynomial_ring"
      }
    ],
    "tag": "0DQMDSVSXJ",
    "uses": [
      "0Q9EZS90GY",
      "0GGE0Z0JQ1",
      "0125QVNCFQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0DQMDSVSXJ/",
  "links": [
    {
      "field": "uses",
      "target": "0Q9EZS90GY"
    },
    {
      "field": "uses",
      "target": "0GGE0Z0JQ1"
    },
    {
      "field": "uses",
      "target": "0125QVNCFQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0DQMDSVSXJ.md",
  "msc": [
    "13F20"
  ],
  "name": "Universal property of the polynomial ring",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DQ/0DQMDSVSXJ.md",
  "status": null,
  "tag": "0DQMDSVSXJ",
  "uri": "m2://0DQMDSVSXJ",
  "url": "https://math2.tamnd.com/wiki/universal-property-of-the-polynomial-ring/",
  "writing": {
    "tag": "0DQMDSVSXJ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}