There is a related question regarding this, but almost all answers lean towards algebraic geometry, which I know none of. The wikipedia page says something related to algebraic number theory, another field I know nothing of.
So my question came up during reading Cohen's theorem which states
A ring $R$ is Noetherian iff every prime ideal of $R$ is finitely generated.
Basically this theorem says we can reduce the study to the prime ideals of $R$ instead of just "regular" ideals of $R$ since we know
A ring $R$ is Noetherian iff every ideal of $R$ is finitely generated.
So what is so fundamental in the role prime ideals play? And I don't mean in this particular case, but in general. Do they generate certain structure/objects in algebra like how primes generate all the other integers? Is there something "easier" with regards to prime ideals? Does reducing theorems to prime ideals give us something more fundamental?
Sorry if these questions sound all the same, I am just looking for some simple answers.