prime ideal

prime ideal (plural prime ideals)

  1. (algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ\subseteq I\implies P\subseteq I or Q\subseteq I.
    • 1960 [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, Volume II, 1975, Springer, page 39 ↗,
      Given a prime number p, there is only a finite number of prime ideals \mathfrak{p} in \mathfrak{o} such that \mathfrak{p}\cap J=p (they are the prime ideals of \mathfrak{o}p).
    • 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), Bartel Leendert 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 \mathfrak{o} on the basis of Axiom II; thus, in that section there are no lower prime ideals but \mathfrak{o}. Since every ideal \mathfrak{a}\ne\mathfrak{o} is divisible by a prime ideal distinct from \mathfrak{o} (proof: from among all the divisors of a distinct from \mathfrak{o} choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to \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 in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab\in P, it follows that a\in P or b\in P; equivalently, P is a prime ideal if and only if the factor ring R/P is a domain.
  2. In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab \in I \implies a \in I or b \in I.
  • Italian: ideale primo

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