"factor ideal" meaning in All languages combined

{
  "forms": [
    {
      "form": "factor ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "factor ideal (plural factor 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": [
        {
          "text": "1965 [Prentice-Hall], Seth Warner, Modern Algebra, 1990, Dover, page 651,\nConsequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A."
        },
        {
          "text": "1985 [W. H. Freeman and Company], Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,\nWe shall now call the sequence of order ideals, ann z₁, ann z₂, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M."
        },
        {
          "ref": "2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246",
          "text": "Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell.[…]\n2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R)."
      ],
      "id": "en-factor_ideal-en-noun-Qq75ptfX",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "coset",
          "coset"
        ],
        [
          "quotient ring",
          "quotient ring"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "factor ideal"
}

{
  "forms": [
    {
      "form": "factor ideals",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "factor ideal (plural factor 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",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "1965 [Prentice-Hall], Seth Warner, Modern Algebra, 1990, Dover, page 651,\nConsequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A."
        },
        {
          "text": "1985 [W. H. Freeman and Company], Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,\nWe shall now call the sequence of order ideals, ann z₁, ann z₂, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M."
        },
        {
          "ref": "2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246",
          "text": "Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell.[…]\n2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "ideal",
          "ideal"
        ],
        [
          "coset",
          "coset"
        ],
        [
          "quotient ring",
          "quotient ring"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "factor ideal"
}