"commutative ring" meaning in English

{
  "forms": [
    {
      "form": "commutative rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "commutative ring (plural commutative rings)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "kind": "topical",
          "langcode": "en",
          "name": "Ring theory",
          "orig": "en:Ring theory",
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            "Mathematics",
            "Formal sciences",
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      ],
      "examples": [
        {
          "ref": "1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129",
          "text": "Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.",
          "type": "quotation"
        },
        {
          "ref": "2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295",
          "text": "As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.",
          "type": "quotation"
        },
        {
          "ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47",
          "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#x5C;inP, it follows that a#x5C;inP or b#x5C;inP; equivalently, P is a prime ideal if and only if the factor ring R#x2F;P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring whose multiplicative operation is commutative."
      ],
      "hyponyms": [
        {
          "sense": "Euclidean domain",
          "word": "field"
        },
        {
          "word": "local ring"
        },
        {
          "sense": "commutative algebra",
          "word": "polynomial ring"
        }
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          "algebra",
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          "ring",
          "ring#Etymology_3"
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        [
          "commutative",
          "commutative#English"
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      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring whose multiplicative operation is commutative."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "a ring whose multiplicative operation is commutative",
          "tags": [
            "masculine"
          ],
          "word": "anello commutativo"
        },
        {
          "alt": "かかんかん",
          "code": "ja",
          "lang": "Japanese",
          "roman": "kakankan",
          "sense": "a ring whose multiplicative operation is commutative",
          "word": "可換環"
        },
        {
          "code": "ro",
          "lang": "Romanian",
          "sense": "a ring whose multiplicative operation is commutative",
          "tags": [
            "neuter"
          ],
          "word": "inel comutativ"
        }
      ],
      "wikipedia": [
        "commutative ring"
      ]
    }
  ],
  "word": "commutative ring"
}

{
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    {
      "form": "commutative rings",
      "tags": [
        "plural"
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    {
      "sense": "Euclidean domain",
      "word": "field"
    },
    {
      "word": "local ring"
    },
    {
      "sense": "commutative algebra",
      "word": "polynomial ring"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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      "examples": [
        {
          "ref": "1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129",
          "text": "Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.",
          "type": "quotation"
        },
        {
          "ref": "2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295",
          "text": "As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.",
          "type": "quotation"
        },
        {
          "ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47",
          "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#x5C;inP, it follows that a#x5C;inP or b#x5C;inP; equivalently, P is a prime ideal if and only if the factor ring R#x2F;P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring whose multiplicative operation is commutative."
      ],
      "links": [
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          "algebra"
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          "ring#Etymology_3"
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        ]
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      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring whose multiplicative operation is commutative."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "commutative ring"
      ]
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "a ring whose multiplicative operation is commutative",
      "tags": [
        "masculine"
      ],
      "word": "anello commutativo"
    },
    {
      "alt": "かかんかん",
      "code": "ja",
      "lang": "Japanese",
      "roman": "kakankan",
      "sense": "a ring whose multiplicative operation is commutative",
      "word": "可換環"
    },
    {
      "code": "ro",
      "lang": "Romanian",
      "sense": "a ring whose multiplicative operation is commutative",
      "tags": [
        "neuter"
      ],
      "word": "inel comutativ"
    }
  ],
  "word": "commutative ring"
}