"reduced ring" meaning in All languages combined

{
  "forms": [
    {
      "form": "reduced rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "reduced ring (plural reduced 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"
        }
      ],
      "examples": [
        {
          "ref": "1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401",
          "text": "We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.",
          "type": "quotation"
        },
        {
          "ref": "2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365",
          "text": "Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#x5C;inM and a#x5C;inR, ma#x3D;0 implies mR#x5C;capMa#x3D;0. Various results of reduced rings are extended to reduced modules.",
          "type": "quotation"
        },
        {
          "ref": "2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210",
          "text": "In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0."
      ],
      "id": "en-reduced_ring-en-noun-ZNJVTz3H",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "nonzero",
          "nonzero"
        ],
        [
          "nilpotent",
          "nilpotent"
        ],
        [
          "element",
          "element"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0."
      ],
      "related": [
        {
          "word": "reduced algebra"
        },
        {
          "word": "reduced scheme"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "ring that has no nonzero nilpotent elements",
          "tags": [
            "masculine"
          ],
          "word": "reduzierter Ring"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "ring that has no nonzero nilpotent elements",
          "tags": [
            "masculine"
          ],
          "word": "anello ridotto"
        }
      ],
      "wikipedia": [
        "reduced ring"
      ]
    }
  ],
  "word": "reduced ring"
}

{
  "forms": [
    {
      "form": "reduced rings",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "reduced ring (plural reduced rings)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "reduced algebra"
    },
    {
      "word": "reduced scheme"
    }
  ],
  "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": [
        {
          "ref": "1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401",
          "text": "We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.",
          "type": "quotation"
        },
        {
          "ref": "2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365",
          "text": "Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#x5C;inM and a#x5C;inR, ma#x3D;0 implies mR#x5C;capMa#x3D;0. Various results of reduced rings are extended to reduced modules.",
          "type": "quotation"
        },
        {
          "ref": "2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210",
          "text": "In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "ring",
          "ring"
        ],
        [
          "nonzero",
          "nonzero"
        ],
        [
          "nilpotent",
          "nilpotent"
        ],
        [
          "element",
          "element"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "reduced ring"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "ring that has no nonzero nilpotent elements",
      "tags": [
        "masculine"
      ],
      "word": "reduzierter Ring"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "ring that has no nonzero nilpotent elements",
      "tags": [
        "masculine"
      ],
      "word": "anello ridotto"
    }
  ],
  "word": "reduced ring"
}