TOPICS
Search

Prime Ideal


A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p is a prime number.

In any principal ideal domain, prime ideals are generated by prime elements. Prime ideals generalize the concept of primality to more general commutative rings.

An ideal I is prime iff the quotient ring R/I is an integral domain because x in I iff x=0 (mod I). Technically, some authors choose not to allow the trivial ring R={0} as a commutative ring, in which case they usually require prime ideals to be proper ideals.

A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, {0} is a prime ideal, as it is in any integral domain. Note that this is the exception to the statement that all prime ideals in the integers are generated by prime numbers. While this might seem silly to allow this case, in some rings the structure of the prime ideals, the Zariski topology, is more interesting. For instance, in polynomials in two variables with complex coefficients C[x,y], the ideals

 <0> subset <(y-x-1)> subset <(x-2),(y-3)>

are all prime.

One consequence of the definition is that the set of elements not in a prime ideal, R-p, is closed under multiplication. This allows one to localize at p by considering the ring of fractions. This ring is analogous to the construction of the rationals as fractions of integers, except that the denominator must be in R-p. The only maximal ideal in this ring is the ideal extension of p.

From the perspective of algebraic geometry, ideals correspond to varieties. Because multiplication corresponds to union (such as xy=0 implies x=0 or y=0), a prime ideal corresponds to an irreducible variety.


See also

Dedekind Ring, Ideal, Irreducible Variety, Krull Dimension, Maximal Ideal, Stickelberger Relation, Stone Space

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Prime Ideal." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PrimeIdeal.html

Subject classifications