"simple algebra" meaning in English

{
  "forms": [
    {
      "form": "simple algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "simple algebra (plural simple algebras)",
      "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"
        }
      ],
      "examples": [
        {
          "text": "1945 [McGraw-Hill], E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,\nA semi-simple algebra over F is either simple or the direct sum of simple algebras over F."
        },
        {
          "text": "1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,\nIf A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ."
        },
        {
          "ref": "2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198",
          "text": "One proves that if #x5C;mathcal#x7B;L#x7D; is a simple algebra over #x5C;mathbb#x7B;R#x7D;, then L#x5F;C is either a simple algebra over #x5C;mathbb#x7B;C#x7D;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #x5C;mathbb#x7B;R#x7D; reduces to a similar problem over #x5C;mathbb#x7B;C#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0)."
      ],
      "hyponyms": [
        {
          "word": "central simple algebra"
        }
      ],
      "id": "en-simple_algebra-en-noun-XxwTaOZH",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "algebra",
          "algebra#English"
        ],
        [
          "nontrivial",
          "nontrivial"
        ],
        [
          "proper",
          "proper#English"
        ],
        [
          "two-sided",
          "two-sided"
        ],
        [
          "ideals",
          "ideal#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0)."
      ],
      "related": [
        {
          "word": "semisimple algebra"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero",
          "tags": [
            "feminine"
          ],
          "word": "algebra semplice"
        }
      ],
      "wikipedia": [
        "simple algebra"
      ]
    }
  ],
  "word": "simple algebra"
}

{
  "forms": [
    {
      "form": "simple algebras",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "simple algebra (plural simple algebras)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "central simple algebra"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "semisimple algebra"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "1945 [McGraw-Hill], E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,\nA semi-simple algebra over F is either simple or the direct sum of simple algebras over F."
        },
        {
          "text": "1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,\nIf A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ."
        },
        {
          "ref": "2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198",
          "text": "One proves that if #x5C;mathcal#x7B;L#x7D; is a simple algebra over #x5C;mathbb#x7B;R#x7D;, then L#x5F;C is either a simple algebra over #x5C;mathbb#x7B;C#x7D;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #x5C;mathbb#x7B;R#x7D; reduces to a similar problem over #x5C;mathbb#x7B;C#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "algebra",
          "algebra#English"
        ],
        [
          "nontrivial",
          "nontrivial"
        ],
        [
          "proper",
          "proper#English"
        ],
        [
          "two-sided",
          "two-sided"
        ],
        [
          "ideals",
          "ideal#English"
        ]
      ],
      "raw_glosses": [
        "(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "simple algebra"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero",
      "tags": [
        "feminine"
      ],
      "word": "algebra semplice"
    }
  ],
  "word": "simple algebra"
}