"minimal ideal" meaning in All languages combined

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    {
      "form": "minimal ideals",
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  "lang_code": "en",
  "pos": "noun",
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      "coordinate_terms": [
        {
          "word": "maximal ideal"
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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": "quote"
        },
        {
          "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.",
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        },
        {
          "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": "quote"
        }
      ],
      "glosses": [
        "A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
      "id": "en-minimal_ideal-en-noun-0PtfZEZI",
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        "(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
      ],
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        {
          "word": "minimal left ideal"
        },
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          "ref": "1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33:",
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        "(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal."
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