"product ring" meaning in English

{
  "forms": [
    {
      "form": "product rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "product ring (plural product 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": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "cup product ring"
        },
        {
          "word": "semidirect product ring"
        }
      ],
      "examples": [
        {
          "ref": "1996, George M. Bergman, Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, page 246",
          "text": "For instance, let X be an infinite set, R the product ring K#x5C;#x21;, and A the set of homomorphisms R#x5C;rightarrowK given by evaluation at all the elements of X. Then the hypothesis of the Lemma holds, but #x5C;text#x7B;Int#x7D;#x5F;A(R) is the product ring k#x5C;#x21;, which is not in general free as a k-module.",
          "type": "quotation"
        },
        {
          "ref": "2003, Erdoğan S. Şuhubi, Functional Analysis, Springer, page 63",
          "text": "Thus the set X#x5C;timesY becomes a ring with these operations and it is called the product ring. The identity element of the product ring with respect to the addition is obviously (0,0) where 0 and 0 are identity elements of addition in the rings X and Y#x5C;#x21;, respectively.",
          "type": "quotation"
        },
        {
          "text": "2007, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique], Springer, page 101,\nThen A_λ is isomorphic to the product ring k×k via the homomorphism sending X to (α,-α)."
        }
      ],
      "glosses": [
        "A ring that is the direct product of rings."
      ],
      "id": "en-product_ring-en-noun-WX7sKz6W",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "direct product",
          "direct product"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(mathematics, ring theory) A ring that is the direct product of rings."
      ],
      "synonyms": [
        {
          "sense": "ring that is the direct product of rings",
          "word": "direct product ring"
        }
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "ring that is the direct product of rings",
          "tags": [
            "masculine"
          ],
          "word": "produit d’anneaux"
        }
      ],
      "wikipedia": [
        "product ring"
      ]
    }
  ],
  "word": "product ring"
}

{
  "derived": [
    {
      "word": "cup product ring"
    },
    {
      "word": "semidirect product ring"
    }
  ],
  "forms": [
    {
      "form": "product rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "product ring (plural product rings)",
      "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:Mathematics"
      ],
      "examples": [
        {
          "ref": "1996, George M. Bergman, Adam O. Hausknecht, Cogroups and Co-rings in Categories of Associative Rings, American Mathematical Society, page 246",
          "text": "For instance, let X be an infinite set, R the product ring K#x5C;#x21;, and A the set of homomorphisms R#x5C;rightarrowK given by evaluation at all the elements of X. Then the hypothesis of the Lemma holds, but #x5C;text#x7B;Int#x7D;#x5F;A(R) is the product ring k#x5C;#x21;, which is not in general free as a k-module.",
          "type": "quotation"
        },
        {
          "ref": "2003, Erdoğan S. Şuhubi, Functional Analysis, Springer, page 63",
          "text": "Thus the set X#x5C;timesY becomes a ring with these operations and it is called the product ring. The identity element of the product ring with respect to the addition is obviously (0,0) where 0 and 0 are identity elements of addition in the rings X and Y#x5C;#x21;, respectively.",
          "type": "quotation"
        },
        {
          "text": "2007, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique], Springer, page 101,\nThen A_λ is isomorphic to the product ring k×k via the homomorphism sending X to (α,-α)."
        }
      ],
      "glosses": [
        "A ring that is the direct product of rings."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "direct product",
          "direct product"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(mathematics, ring theory) A ring that is the direct product of rings."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "product ring"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "ring that is the direct product of rings",
      "word": "direct product ring"
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "ring that is the direct product of rings",
      "tags": [
        "masculine"
      ],
      "word": "produit d’anneaux"
    }
  ],
  "word": "product ring"
}