"minimal ideal" meaning in All languages combined

{
  "forms": [
    {
      "form": "minimal ideals",
      "tags": [
        "plural"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
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          "name": "Algebra",
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      "coordinate_terms": [
        {
          "word": "maximal ideal"
        }
      ],
      "examples": [
        {
          "ref": "1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii",
          "text": "Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.",
          "type": "quotation"
        },
        {
          "ref": "1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33",
          "text": "Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.",
          "type": "quotation"
        },
        {
          "ref": "2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596",
          "text": "Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
      "id": "en-minimal_ideal-en-noun-0PtfZEZI",
      "links": [
        [
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        [
          "nonzero",
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        [
          "two-sided",
          "two-sided"
        ],
        [
          "ideal",
          "ideal"
        ]
      ],
      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
      "related": [
        {
          "word": "minimal left ideal"
        },
        {
          "word": "minimal right ideal"
        }
      ],
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        "mathematics",
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      "wikipedia": [
        "minimal ideal"
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  "word": "minimal ideal"
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{
  "coordinate_terms": [
    {
      "word": "maximal ideal"
    }
  ],
  "forms": [
    {
      "form": "minimal ideals",
      "tags": [
        "plural"
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    }
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  "head_templates": [
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  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "minimal left ideal"
    },
    {
      "word": "minimal right ideal"
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  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
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      "examples": [
        {
          "ref": "1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii",
          "text": "Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.",
          "type": "quotation"
        },
        {
          "ref": "1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33",
          "text": "Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.",
          "type": "quotation"
        },
        {
          "ref": "2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596",
          "text": "Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
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      "qualifier": "ring theory",
      "raw_glosses": [
        "(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
      "topics": [
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      ],
      "wikipedia": [
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  "word": "minimal ideal"
}