Prime ideal

The set of prime ideals
The set of prime ideals in a commutative unital ring is termed its spectrum. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.

Stronger properties

 * Weaker than::Maximal ideal
 * Weaker than::Minimal prime ideal
 * Weaker than::Principal prime ideal

Weaker properties

 * Stronger than::Irreducible ideal
 * Stronger than::Primary ideal
 * Stronger than::Radical ideal
 * Stronger than::Ideal with prime radical

Metaproperties
An intersection of prime ideals need not be prime. In fact, an ideal is an intersection of prime ideals iff it is a radical ideal.

If $$f:R \to S$$ is a homomorphism of commutative unital rings, and $$I$$ is a prime ideal of $$S$$, then the contraction of $$I$$ to $$R$$, denoted $$I^c$$, (same as $$f^{-1}(I)$$) is a prime ideal in $$R$$.

Note that this implies the intermediate subring condition and the transfer condition on ideals.

If an ideal is prime in the whole ring, it is also prime in any intermediate subring. This is related to the fact that any subring of an integral domain is an integral domain.

If $$I$$ is a prime ideal in $$R$$, and $$S$$ is any subring of $$R$$, then $$I \cap S$$ is a prime ideal in $$S$$. Note that this implies the intermediate ring condition as well.

Effect of property operators
An ideal in a commutative unital ring is expressible as an intersection of prime ideals iff it is a radical ideal.

In rings of integers
A ring of integers in a number field is a Dedekind domain, hence any nonzero prime ideal in this ring is a maximal ideal.