"polynomial ring" meaning in English

{
  "forms": [
    {
      "form": "polynomial rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "polynomial ring (plural polynomial 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": [
        {
          "english": "= Ore extension",
          "word": "skew polynomial ring"
        }
      ],
      "examples": [
        {
          "ref": "1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270",
          "text": "It then follows that if A is a graded ring over a local ring, A is a homomorphic image of a polynomial ring over a regular local ring. For the sake of brevity, we refer to a graded polynomial ring over a regular local ring simply as a graded polynomial ring.",
          "type": "quotation"
        },
        {
          "ref": "2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106",
          "text": "In Section 3.2 we shall study the special properties of a polynomial ring over a field; for the moment we note a property of polynomial rings which applies quite generally, the Hilbert basis theorem (after David Hilbert, 1862-1943):\nTheorem 3.3\nIf R is any right Noetherian ring, the polynomial ring R#x5B;x#x5D; is again right Noetherian.",
          "type": "quotation"
        },
        {
          "ref": "2009, Jesse Elliott, “Some new approaches to integer-valued polynomial rings”, in Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, editors, Commutative Algebra and Its Applications: Proceedings of the 5th International Fez Conference, Walter de Gruyter, page 223",
          "text": "Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K."
      ],
      "hypernyms": [
        {
          "word": "associative algebra"
        },
        {
          "word": "commutative algebra"
        },
        {
          "word": "commutative ring"
        }
      ],
      "id": "en-polynomial_ring-en-noun-AjTqHLHh",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "commutative algebra",
          "commutative algebra"
        ],
        [
          "polynomials",
          "polynomial#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K."
      ],
      "related": [
        {
          "word": "ring of formal power series"
        },
        {
          "word": "ring of polynomial functions"
        },
        {
          "word": "ring of regular functions"
        }
      ],
      "synonyms": [
        {
          "sense": "ring formed from polynomials",
          "word": "polynomial algebra"
        },
        {
          "sense": "ring formed from polynomials",
          "word": "ring of polynomials"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "ring formed from polynomials",
          "tags": [
            "masculine"
          ],
          "word": "anello dei polinomi"
        }
      ],
      "wikipedia": [
        "polynomial ring"
      ]
    }
  ],
  "word": "polynomial ring"
}

{
  "derived": [
    {
      "english": "= Ore extension",
      "word": "skew polynomial ring"
    }
  ],
  "forms": [
    {
      "form": "polynomial rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "polynomial ring (plural polynomial rings)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "associative algebra"
    },
    {
      "word": "commutative algebra"
    },
    {
      "word": "commutative ring"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "ring of formal power series"
    },
    {
      "word": "ring of polynomial functions"
    },
    {
      "word": "ring of regular functions"
    }
  ],
  "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": "1998, Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, page 270",
          "text": "It then follows that if A is a graded ring over a local ring, A is a homomorphic image of a polynomial ring over a regular local ring. For the sake of brevity, we refer to a graded polynomial ring over a regular local ring simply as a graded polynomial ring.",
          "type": "quotation"
        },
        {
          "ref": "2000, Paul M. Cohn, Introduction to Ring Theory, Springer, page 106",
          "text": "In Section 3.2 we shall study the special properties of a polynomial ring over a field; for the moment we note a property of polynomial rings which applies quite generally, the Hilbert basis theorem (after David Hilbert, 1862-1943):\nTheorem 3.3\nIf R is any right Noetherian ring, the polynomial ring R#x5B;x#x5D; is again right Noetherian.",
          "type": "quotation"
        },
        {
          "ref": "2009, Jesse Elliott, “Some new approaches to integer-valued polynomial rings”, in Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson, editors, Commutative Algebra and Its Applications: Proceedings of the 5th International Fez Conference, Walter de Gruyter, page 223",
          "text": "Because they possess a rich theory and provide an excellent source of examples and counterexamples, integer-valued polynomial rings have attained some prominence in the theory of non-Noetherian commutative rings.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "commutative algebra",
          "commutative algebra"
        ],
        [
          "polynomials",
          "polynomial#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "polynomial ring"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "ring formed from polynomials",
      "word": "polynomial algebra"
    },
    {
      "sense": "ring formed from polynomials",
      "word": "ring of polynomials"
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "ring formed from polynomials",
      "tags": [
        "masculine"
      ],
      "word": "anello dei polinomi"
    }
  ],
  "word": "polynomial ring"
}