"prime ideal" meaning in All languages combined

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  "etymology_text": "By analogy with the notion of prime number in number theory.",
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        "plural"
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  "lang_code": "en",
  "pos": "noun",
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        {
          "ref": "1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39",
          "text": "Given a prime number p, there is only a finite number of prime ideals #x5C;mathfrak#x7B;p#x7D; in #x5C;mathfrak#x7B;o#x7D; such that #x5C;mathfrak#x7B;p#x7D;#x5C;capJ#x3D;p (they are the prime ideals of #x5C;mathfrak#x7B;o#x7D;p).",
          "type": "quotation"
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,\nIn the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o."
        },
        {
          "ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47",
          "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#x5C;inP, it follows that a#x5C;inP or b#x5C;inP; equivalently, P is a prime ideal if and only if the factor ring R#x2F;P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I."
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        "(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I."
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      "glosses": [
        "In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab∈I⟹a∈I or b∈I."
      ],
      "id": "en-prime_ideal-en-noun-j2vj06BV",
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        [
          "commutative ring",
          "commutative ring"
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      "translations": [
        {
          "_dis1": "42 58",
          "code": "it",
          "lang": "Italian",
          "sense": "(ring theory) type of ideal",
          "tags": [
            "masculine"
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          "word": "ideale primo"
        }
      ]
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  "wikipedia": [
    "prime ideal"
  ],
  "word": "prime ideal"
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  "etymology_text": "By analogy with the notion of prime number in number theory.",
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      "examples": [
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          "ref": "1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39",
          "text": "Given a prime number p, there is only a finite number of prime ideals #x5C;mathfrak#x7B;p#x7D; in #x5C;mathfrak#x7B;o#x7D; such that #x5C;mathfrak#x7B;p#x7D;#x5C;capJ#x3D;p (they are the prime ideals of #x5C;mathfrak#x7B;o#x7D;p).",
          "type": "quotation"
        },
        {
          "text": "1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,\nIn the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o."
        },
        {
          "ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47",
          "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#x5C;inP, it follows that a#x5C;inP or b#x5C;inP; equivalently, P is a prime ideal if and only if the factor ring R#x2F;P is a domain.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I."
      ],
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        "(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I."
      ],
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        "In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab∈I⟹a∈I or b∈I."
      ],
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          "commutative ring",
          "commutative ring"
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "(ring theory) type of ideal",
      "tags": [
        "masculine"
      ],
      "word": "ideale primo"
    }
  ],
  "wikipedia": [
    "prime ideal"
  ],
  "word": "prime ideal"
}