prime ideal

plural prime ideals

1. ring theoryalgebraAny (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.

   Given a prime number p, there is only a finite number of prime ideals #92;mathfrak#123;p#125; in #92;mathfrak#123;o#125; such that #92;mathfrak#123;p#125;#92;capJ#61;p (they are the prime ideals of #92;mathfrak#123;o#125;p).

   1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39

   2 more examples

   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 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.

   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#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

   2004, K. R. Goodearl; R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47

2. 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.