"unique factorization ring" meaning in English

{
  "forms": [
    {
      "form": "unique factorization rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "unique factorization ring (plural unique factorization 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"
        }
      ],
      "glosses": [
        "A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes."
      ],
      "hyponyms": [
        {
          "word": "unique factorization domain"
        }
      ],
      "id": "en-unique_factorization_ring-en-noun-lcXzSR6O",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "unique factorization ring"
}

{
  "forms": [
    {
      "form": "unique factorization rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "unique factorization ring (plural unique factorization rings)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "unique factorization domain"
    }
  ],
  "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",
        "en:Algebra"
      ],
      "glosses": [
        "A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "unique factorization ring"
}