Why are prime ideals crucial in algebraic geometry?

[Metric_Space]

Homework Statement

Why are prime ideals so important?

Homework Equations

The Attempt at a Solution

 

[micromass]

Hi Metric_Space

1) Because they form a generalization for prime numbers, and prime numbers are important.
2) Because they correspond to irreducible curves in geometry. For example, the curve y=x² is irreducible, and indeed, the ideal (y-x²) is prime in \mathbb{C}[x,y]. On the other, the curve xy=0 is not irreducible since it exists out of the pieces x=0 and y=0. And indeed, the ideal (xy) is not prime.