"simple ring" meaning in All languages combined

{
  "forms": [
    {
      "form": "simple rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "simple ring (plural simple 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 French translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Italian translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:",
          "text": "Theorem 2. A semi-simple ring #x5C;mathfrak#x7B;A#x7D; satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.",
          "type": "quote"
        },
        {
          "ref": "1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:",
          "text": "By theorem 7.7.1 any field is a simple ring.",
          "type": "quote"
        },
        {
          "ref": "1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:",
          "text": "A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is Fₙ, the n × n matrix ring over a field F, n > 1.",
          "type": "quote"
        },
        {
          "ref": "2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:",
          "text": "Since M#x5F;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#x5F;n(D) is a left simple ring. We shall see that every simple ring is isomorphic to M#x5F;n(D) for some n, and for some division ring D.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself)."
      ],
      "hypernyms": [
        {
          "word": "local ring"
        }
      ],
      "hyponyms": [
        {
          "word": "field"
        }
      ],
      "id": "en-simple_ring-en-noun-n4-Gb7cr",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "nontrivial",
          "nontrivial"
        ],
        [
          "two-sided",
          "two-sided"
        ],
        [
          "ideals",
          "ideal#English"
        ],
        [
          "zero",
          "zero#English"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself)."
      ],
      "related": [
        {
          "word": "semisimple ring"
        },
        {
          "word": "simple module"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "ring that contains no ideals other than the zero ideal and the ring itself",
          "tags": [
            "masculine"
          ],
          "word": "anneau simple"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "ring that contains no ideals other than the zero ideal and the ring itself",
          "tags": [
            "masculine"
          ],
          "word": "anello semplice"
        }
      ],
      "wikipedia": [
        "simple ring"
      ]
    }
  ],
  "word": "simple ring"
}

{
  "forms": [
    {
      "form": "simple rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "simple ring (plural simple rings)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "local ring"
    }
  ],
  "hyponyms": [
    {
      "word": "field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "semisimple ring"
    },
    {
      "word": "simple module"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with French translations",
        "Terms with Italian translations",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:",
          "text": "Theorem 2. A semi-simple ring #x5C;mathfrak#x7B;A#x7D; satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.",
          "type": "quote"
        },
        {
          "ref": "1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:",
          "text": "By theorem 7.7.1 any field is a simple ring.",
          "type": "quote"
        },
        {
          "ref": "1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:",
          "text": "A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is Fₙ, the n × n matrix ring over a field F, n > 1.",
          "type": "quote"
        },
        {
          "ref": "2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:",
          "text": "Since M#x5F;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#x5F;n(D) is a left simple ring. We shall see that every simple ring is isomorphic to M#x5F;n(D) for some n, and for some division ring D.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring#English"
        ],
        [
          "nontrivial",
          "nontrivial"
        ],
        [
          "two-sided",
          "two-sided"
        ],
        [
          "ideals",
          "ideal#English"
        ],
        [
          "zero",
          "zero#English"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "simple ring"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "ring that contains no ideals other than the zero ideal and the ring itself",
      "tags": [
        "masculine"
      ],
      "word": "anneau simple"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "ring that contains no ideals other than the zero ideal and the ring itself",
      "tags": [
        "masculine"
      ],
      "word": "anello semplice"
    }
  ],
  "word": "simple ring"
}