Jacobson radical in English

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{
  "forms": [
    {
      "form": "Jacobson radicals",
      "tags": [
        "plural"
      ]
    }
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  "head_templates": [
    {
      "args": {},
      "expansion": "Jacobson radical (plural Jacobson radicals)",
      "name": "en-noun"
    }
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "name": "English entries with incorrect language header",
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      "glosses": [
        "The operation that produces (from a given ring) the ideal of elements which annihilates all simple modules (over the given ring); the result of this operation. The Jacobson radical of a ring R is denoted J(R)."
      ],
      "id": "en-Jacobson_radical-en-noun-ypoByrNB",
      "links": [
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          "mathematics"
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        [
          "ring theory",
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        ],
        [
          "ideal",
          "ideal"
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          "element",
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          "annihilate",
          "annihilate"
        ],
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          "simple",
          "simple"
        ],
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          "module",
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        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(mathematics, ring theory) The operation that produces (from a given ring) the ideal of elements which annihilates all simple modules (over the given ring); the result of this operation. The Jacobson radical of a ring R is denoted J(R)."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "Jacobson radical"
}

[Show JSON for raw wiktextract data ▼] [Hide JSON for raw wiktextract data ▲]

{
  "forms": [
    {
      "form": "Jacobson radicals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Jacobson radical (plural Jacobson radicals)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
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        "Pages with entries",
        "en:Mathematics"
      ],
      "glosses": [
        "The operation that produces (from a given ring) the ideal of elements which annihilates all simple modules (over the given ring); the result of this operation. The Jacobson radical of a ring R is denoted J(R)."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "ring theory",
          "ring theory"
        ],
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          "operation",
          "operation"
        ],
        [
          "ideal",
          "ideal"
        ],
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          "element"
        ],
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          "annihilate",
          "annihilate"
        ],
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          "simple",
          "simple"
        ],
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        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(mathematics, ring theory) The operation that produces (from a given ring) the ideal of elements which annihilates all simple modules (over the given ring); the result of this operation. The Jacobson radical of a ring R is denoted J(R)."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "Jacobson radical"
}

This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.

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"Jacobson radical" meaning in English

Noun