"quotient ring" meaning in English

{
  "forms": [
    {
      "form": "quotient rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quotient ring (plural quotient rings)",
      "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"
        }
      ],
      "derived": [
        {
          "word": "left quotient ring"
        },
        {
          "word": "right quotient ring"
        },
        {
          "word": "maximal quotient ring"
        }
      ],
      "examples": [
        {
          "ref": "1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39",
          "text": "The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding maximal right and left quotient rings.",
          "type": "quotation"
        },
        {
          "ref": "2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42",
          "text": "On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.",
          "type": "quotation"
        },
        {
          "ref": "2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10",
          "text": "2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R."
      ],
      "hyponyms": [
        {
          "word": "residue field"
        }
      ],
      "id": "en-quotient_ring-en-noun-eoj4r1YU",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "ideal",
          "ideal#English"
        ],
        [
          "coset",
          "coset"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R."
      ],
      "related": [
        {
          "word": "quotient group"
        },
        {
          "word": "quotient module"
        },
        {
          "word": "quotient object"
        },
        {
          "word": "quotient space"
        }
      ],
      "synonyms": [
        {
          "sense": "ring whose elements are the cosets of an ideal",
          "word": "difference ring"
        },
        {
          "sense": "ring whose elements are the cosets of an ideal",
          "word": "factor ring"
        },
        {
          "sense": "ring whose elements are the cosets of an ideal",
          "word": "residue class ring"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "ring whose elements are the cosets of an ideal",
          "tags": [
            "masculine"
          ],
          "word": "anneau quotient"
        }
      ],
      "wikipedia": [
        "quotient ring"
      ]
    }
  ],
  "word": "quotient ring"
}

{
  "derived": [
    {
      "word": "left quotient ring"
    },
    {
      "word": "right quotient ring"
    },
    {
      "word": "maximal quotient ring"
    }
  ],
  "forms": [
    {
      "form": "quotient rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quotient ring (plural quotient rings)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "residue field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "quotient group"
    },
    {
      "word": "quotient module"
    },
    {
      "word": "quotient object"
    },
    {
      "word": "quotient space"
    }
  ],
  "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": "1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39",
          "text": "The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding maximal right and left quotient rings.",
          "type": "quotation"
        },
        {
          "ref": "2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42",
          "text": "On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.",
          "type": "quotation"
        },
        {
          "ref": "2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10",
          "text": "2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "ideal",
          "ideal#English"
        ],
        [
          "coset",
          "coset"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "quotient ring"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "ring whose elements are the cosets of an ideal",
      "word": "difference ring"
    },
    {
      "sense": "ring whose elements are the cosets of an ideal",
      "word": "factor ring"
    },
    {
      "sense": "ring whose elements are the cosets of an ideal",
      "word": "residue class ring"
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "ring whose elements are the cosets of an ideal",
      "tags": [
        "masculine"
      ],
      "word": "anneau quotient"
    }
  ],
  "word": "quotient ring"
}