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prime ideal

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prime ideal

名詞

prime ideal (複数形 prime ideals)

(algebra, ring theory) Any (two-sided) ideal I {\displaystyle I} such that for arbitrary ideals P {\displaystyle P} and Q {\displaystyle Q} , P Q ⊆ I ⟹ P ⊆ I {\displaystyle PQ\subseteq I\implies P\subseteq I} or Q ⊆ I {\displaystyle Q\subseteq I} .
- 1960 [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, Volume II, 1975, Springer, page 39,
  Given a prime number p {\displaystyle p} $p$ , there is only a finite number of prime ideals p {\displaystyle {\mathfrak {p}}} ${\mathfrak {p}}$ in o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ such that p ∩ J = p {\displaystyle {\mathfrak {p}}\cap J=p} ${\mathfrak {p}}\cap J=p$ (they are the prime ideals of o p {\displaystyle {\mathfrak {o}}p} ${\mathfrak {o}}p$ ).
- 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,
  In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ on the basis of Axiom II; thus, in that section there are no lower prime ideals but o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ . Since every ideal a ≠ o {\displaystyle {\mathfrak {a}}\neq {\mathfrak {o}}} ${\mathfrak {a}}\neq {\mathfrak {o}}$ is divisible by a prime ideal distinct from o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ (proof: from among all the divisors of a distinct from o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o {\displaystyle {\mathfrak {o}}} ${\mathfrak {o}}$ .
- 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd Edition, Cambridge University Press, page 47,
  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 {\displaystyle P} $P$ in a commutative ring R {\displaystyle R} $R$ is prime if, whenever we have two elements a {\displaystyle a} $a$ and b {\displaystyle b} $b$ of R {\displaystyle R} $R$ such that a b ∈ P {\displaystyle ab\in P} $ab\in P$ , it follows that a ∈ P {\displaystyle a\in P} $a\in P$ or b ∈ P {\displaystyle b\in P} $b\in P$ ; equivalently, P {\displaystyle P} $P$ is a prime ideal if and only if the factor ring R / P {\displaystyle R/P} $R/P$ is a domain.