"primary ideal" meaning in English

{
  "forms": [
    {
      "form": "primary ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "primary ideal (plural primary 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": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10",
          "text": "The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.",
          "type": "quotation"
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,\nThus all higher primary ideals are symbolic powers of higher prime ideals.\nPrüfer has called the ideals a with the property a* = a v-ideals. The integral v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur."
        },
        {
          "ref": "1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71",
          "text": "A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0."
      ],
      "hyponyms": [
        {
          "sense": "ring theory",
          "word": "prime ideal"
        }
      ],
      "id": "en-primary_ideal-en-noun-0UmJReK9",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "commutative ring",
          "commutative ring"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "integer",
          "integer"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "(ring theory)",
          "tags": [
            "neuter"
          ],
          "word": "primäres Ideal"
        }
      ],
      "wikipedia": [
        "primary ideal"
      ]
    }
  ],
  "word": "primary ideal"
}

{
  "forms": [
    {
      "form": "primary ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "primary ideal (plural primary ideals)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "sense": "ring theory",
      "word": "prime ideal"
    }
  ],
  "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",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10",
          "text": "The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.",
          "type": "quotation"
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,\nThus all higher primary ideals are symbolic powers of higher prime ideals.\nPrüfer has called the ideals a with the property a* = a v-ideals. The integral v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur."
        },
        {
          "ref": "1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71",
          "text": "A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "commutative ring",
          "commutative ring"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "integer",
          "integer"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "primary ideal"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "(ring theory)",
      "tags": [
        "neuter"
      ],
      "word": "primäres Ideal"
    }
  ],
  "word": "primary ideal"
}