"group ring" meaning in All languages combined

{
  "forms": [
    {
      "form": "group rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "group ring (plural group 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": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Portuguese translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Ring theory",
          "orig": "en:Ring theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "glosses": [
        "Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k."
      ],
      "hypernyms": [
        {
          "word": "algebra over a ring"
        },
        {
          "word": "free module"
        },
        {
          "word": "ring"
        }
      ],
      "id": "en-group_ring-en-noun-I2RdT1cI",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "identity",
          "identity"
        ],
        [
          "group",
          "group"
        ]
      ],
      "raw_glosses": [
        "(algebra) Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k."
      ],
      "synonyms": [
        {
          "word": "group algebra"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "pt",
          "lang": "Portuguese",
          "sense": "Translations",
          "tags": [
            "masculine"
          ],
          "word": "anel de grupo"
        }
      ],
      "wikipedia": [
        "group ring"
      ]
    }
  ],
  "word": "group ring"
}

{
  "forms": [
    {
      "form": "group rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "group ring (plural group rings)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "algebra over a ring"
    },
    {
      "word": "free module"
    },
    {
      "word": "ring"
    }
  ],
  "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",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with Portuguese translations",
        "Translation table header lacks gloss",
        "en:Algebra",
        "en:Ring theory"
      ],
      "glosses": [
        "Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "identity",
          "identity"
        ],
        [
          "group",
          "group"
        ]
      ],
      "raw_glosses": [
        "(algebra) Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "group ring"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "group algebra"
    }
  ],
  "translations": [
    {
      "code": "pt",
      "lang": "Portuguese",
      "sense": "Translations",
      "tags": [
        "masculine"
      ],
      "word": "anel de grupo"
    }
  ],
  "word": "group ring"
}