{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2C8JN6XBWY"
    },
    {
      "field": "specializes",
      "target": "49Y9Q949PK"
    }
  ],
  "body": "\n## Summary\n\nA semiperfect ring is a [[semilocal ring|ZDCDS6EWGX]] in which idempotents [[lift|2SRDN8EJG3]] modulo the Jacobson radical.\nEquivalently, every finitely generated left module has a [[projective cover|WGYXR97NPK]].\nEquivalently again, $1$ is a sum of orthogonal idempotents $e_1, \\ldots, e_n$ such that each corner ring $e_i R e_i$ is local.\nLeft Artinian rings, [[local rings|MZ02BKPHJA]] and [[perfect rings|49Y9Q949PK]] are semiperfect.\n\n## Explanations\n\n### L1 human\n\nA semiperfect ring splits, as a module over itself, into finitely many indecomposable pieces, each of which behaves like a local ring.\nThis makes its module theory close to that of finite-dimensional algebras.\nThe ring of $p$-adic integers is semiperfect, being a complete local ring.\nThe ring of fractions $a/b$ with $b$ prime to $6$ is semilocal but not semiperfect.\nModulo its radical it is $\\mathbb{F}_2 \\times \\mathbb{F}_3$, which has idempotents that do not come from the ring.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16L30"
    ],
    "name": "Semiperfect ring",
    "sources": [
      {
        "locator": "Semiperfect ring",
        "url": "https://en.wikipedia.org/wiki/Perfect_ring"
      }
    ],
    "specializes": [
      "ZDCDS6EWGX"
    ],
    "tag": "ZRC2S3ECFK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/ZRC2S3ECFK/",
  "links": [
    {
      "field": "specializes",
      "target": "ZDCDS6EWGX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZRC2S3ECFK.md",
  "msc": [
    "16L30"
  ],
  "name": "Semiperfect ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZRC/ZRC2S3ECFK.md",
  "status": null,
  "tag": "ZRC2S3ECFK",
  "uri": "m2://ZRC2S3ECFK",
  "url": "https://tamnd.github.io/math2/wiki/semiperfect-ring/",
  "writing": {
    "tag": "ZRC2S3ECFK",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}