"radical ideal" meaning in All languages combined

{
  "forms": [
    {
      "form": "radical ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "radical ideal (plural radical 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"
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          "source": "w"
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
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        }
      ],
      "examples": [
        {
          "text": "1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,\nAny intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime."
        },
        {
          "ref": "1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33:",
          "text": "It follows at once that R#x2F;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.",
          "type": "quote"
        },
        {
          "text": "1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,\nThus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] ."
        }
      ],
      "glosses": [
        "An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I)."
      ],
      "hyponyms": [
        {
          "sense": "ideal that is its own radical",
          "word": "vanishing ideal"
        }
      ],
      "id": "en-radical_ideal-en-noun-7i~tRmPX",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ideal",
          "ideal#English"
        ],
        [
          "radical",
          "radical#English"
        ],
        [
          "positive",
          "positive#English"
        ],
        [
          "integer",
          "integer#English"
        ]
      ],
      "qualifier": "commutative algebra; ring theory; commutative algebra; ring theory",
      "raw_glosses": [
        "(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I)."
      ],
      "synonyms": [
        {
          "sense": "ideal that is its own radical",
          "word": "semiprime ideal"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "radical ideal"
}

{
  "forms": [
    {
      "form": "radical ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "radical ideal (plural radical ideals)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "sense": "ideal that is its own radical",
      "word": "vanishing ideal"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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        "English countable nouns",
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        "English terms with quotations",
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      "examples": [
        {
          "text": "1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,\nAny intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime."
        },
        {
          "ref": "1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33:",
          "text": "It follows at once that R#x2F;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.",
          "type": "quote"
        },
        {
          "text": "1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,\nThus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] ."
        }
      ],
      "glosses": [
        "An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ideal",
          "ideal#English"
        ],
        [
          "radical",
          "radical#English"
        ],
        [
          "positive",
          "positive#English"
        ],
        [
          "integer",
          "integer#English"
        ]
      ],
      "qualifier": "commutative algebra; ring theory; commutative algebra; ring theory",
      "raw_glosses": [
        "(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "ideal that is its own radical",
      "word": "semiprime ideal"
    }
  ],
  "word": "radical ideal"
}