prime ideal

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English

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Etymology

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By analogy with the notion of prime number in number theory.

Noun

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prime ideal (plural prime ideals)

  1. (algebra, ring theory) Any (two-sided) ideal such that for arbitrary ideals and , or .
    • 1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:
      Given a prime number , there is only a finite number of prime ideals in such that (they are the prime ideals of ).
    • 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 on the basis of Axiom II; thus, in that section there are no lower prime ideals but . Since every ideal is divisible by a prime ideal distinct from (proof: from among all the divisors of a distinct from choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to .
    • 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 in a commutative ring is prime if, whenever we have two elements and of such that , it follows that or ; equivalently, is a prime ideal if and only if the factor ring is a domain.
  2. In a commutative ring, a (two-sided) ideal such that for arbitrary ring elements and , or .

Translations

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