principal ideal in English

[Show JSON for postprocessed kaikki.org data shown on this page ▼] [Hide JSON for postprocessed kaikki.org data shown on this page ▲]

{
  "forms": [
    {
      "form": "principal ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "principal ideal (plural principal ideals)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "left principal ideal"
        },
        {
          "word": "principal fractional ideal"
        },
        {
          "word": "principal ideal domain"
        },
        {
          "word": "right principal ideal"
        }
      ],
      "glosses": [
        "An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a."
      ],
      "id": "en-principal_ideal-en-noun-u2iAcL3G",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "semigroup",
          "semigroup"
        ],
        [
          "lattice",
          "lattice"
        ],
        [
          "generated",
          "generated"
        ]
      ],
      "raw_glosses": [
        "(algebra) An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a."
      ],
      "synonyms": [
        {
          "sense": "smallest ideal that contains a given element",
          "word": "two-sided principal ideal"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "principal ideal"
      ]
    }
  ],
  "word": "principal ideal"
}

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

{
  "derived": [
    {
      "word": "left principal ideal"
    },
    {
      "word": "principal fractional ideal"
    },
    {
      "word": "principal ideal domain"
    },
    {
      "word": "right principal ideal"
    }
  ],
  "forms": [
    {
      "form": "principal ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "principal ideal (plural principal ideals)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Algebra"
      ],
      "glosses": [
        "An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "semigroup",
          "semigroup"
        ],
        [
          "lattice",
          "lattice"
        ],
        [
          "generated",
          "generated"
        ]
      ],
      "raw_glosses": [
        "(algebra) An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "principal ideal"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "smallest ideal that contains a given element",
      "word": "two-sided principal ideal"
    }
  ],
  "word": "principal ideal"
}

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.

If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.

"principal ideal" meaning in English

Noun

Inflected forms