Prime Ideal

A prime ideal is a proper ideal in a commutative ring with identity such that if , then either or . For example, in the integers, the ideal (i.e., the multiples of ) is prime whenever is a prime number.

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

An ideal is prime iff the quotient ring is an integral domain because iff . Here an integral domain is required to have , consistent with the proper ideal condition.

A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, 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 , the ideals

are all prime.

One consequence of the definition is that the set of elements not in a prime ideal, , is closed under multiplication. This allows one to localize at 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 . The only maximal ideal in this ring is the ideal extension of .

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