{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3PNJYFPJ3P"
    }
  ],
  "body": "\n## Summary\n\nA homogeneous ideal of a [[graded ring|8KB4SQGC0C]] is an [[ideal|096Z9MP2ME]] that contains all homogeneous components of each of its elements.\nEquivalently, it is generated by homogeneous elements.\nHomogeneous ideals of a polynomial ring are the ideals that define subsets of projective space.\nThe quotient by a homogeneous ideal is again graded.\n\n## Explanations\n\n### L1 human\n\nA polynomial is a sum of homogeneous parts of different degrees.\nAn ideal is homogeneous when it never mixes degrees: whenever it contains a polynomial, it contains each homogeneous part separately.\nThe ideal generated by $x^2 - yz$ in $k[x, y, z]$ is homogeneous, while the ideal generated by $x - 1$ in $k[x]$ is not, since it contains $x - 1$ but not $x$.\nThe [[definition|3PNJYFPJ3P]] makes this precise.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13A02",
      "13A15"
    ],
    "name": "Homogeneous ideal",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Graded_ring"
      }
    ],
    "specializes": [
      "096Z9MP2ME"
    ],
    "tag": "6ZBNGS0P98"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/6ZBNGS0P98/",
  "links": [
    {
      "field": "specializes",
      "target": "096Z9MP2ME"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/6ZBNGS0P98.md",
  "msc": [
    "13A02",
    "13A15"
  ],
  "name": "Homogeneous ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6ZB/6ZBNGS0P98.md",
  "status": null,
  "tag": "6ZBNGS0P98",
  "uri": "m2://6ZBNGS0P98",
  "url": "https://math2.tamnd.com/wiki/homogeneous-ideal/",
  "writing": {
    "tag": "6ZBNGS0P98",
    "kind": "concept",
    "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
      }
    ]
  }
}