{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nLet $\\Delta_A = \\set{(a, a) : a \\in A}$ and $\\nabla_A = A \\times A$.\nThe algebra $\\mathbf{A}$ is *simple* when $A$ has at least two elements and its only [[congruences|DCSXXT4SER]] are $\\Delta_A$ and $\\nabla_A$.\n\n## Remarks\n\nEquivalently, the [[congruence lattice|REE07CSXHR]] of $\\mathbf{A}$ is the two-element chain.\nBy the correspondence theorem, a quotient $\\mathbf{A}/\\theta$ is simple exactly when $\\theta$ is a maximal element of $\\operatorname{Con} \\mathbf{A} \\setminus \\set{\\nabla_A}$.\n\nExample: the group $\\Z/p\\Z$ with $p$ prime is simple.\nIts congruences come from its subgroups, and by Lagrange's theorem these are only $\\set{0}$ and $\\Z/p\\Z$.\n\nNon-example: the ring $\\Z$ is not simple.\nCongruence modulo $2$ is a congruence, and it is neither $\\Delta_\\Z$, since $(0, 2)$ belongs to it, nor $\\nabla_\\Z$, since $(0, 1)$ does not.\n\nA field, viewed as a ring, is simple, since its only ideals are $\\set{0}$ and the whole field.\nThe one-element algebra is excluded, because there $\\Delta_A = \\nabla_A$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9CJCXNYNK8",
    "kind": "definition",
    "msc": [
      "08A30",
      "08A05"
    ],
    "name": "Simple algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simple_algebra_(universal_algebra)"
      }
    ],
    "tag": "7KK111QGBZ",
    "uses": [
      "QJGPT2W0FT",
      "DCSXXT4SER",
      "REE07CSXHR"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/7KK111QGBZ/",
  "links": [
    {
      "field": "defines",
      "target": "9CJCXNYNK8"
    },
    {
      "field": "uses",
      "target": "QJGPT2W0FT"
    },
    {
      "field": "uses",
      "target": "DCSXXT4SER"
    },
    {
      "field": "uses",
      "target": "REE07CSXHR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/7KK111QGBZ.md",
  "msc": [
    "08A30",
    "08A05"
  ],
  "name": "Simple algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7KK/7KK111QGBZ.md",
  "status": null,
  "tag": "7KK111QGBZ",
  "uri": "m2://7KK111QGBZ",
  "url": "https://tamnd.github.io/math2/wiki/simple-algebra-definition/",
  "writing": {
    "tag": "7KK111QGBZ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}