TITLE: Motivation behind the definition of Prime Ideal
QUESTION [16 upvotes]: Can someone explain what's the motivation behind the definition of a prime ideal? 
Or why is it exactly called a prime ideal?
Has it anything to do it prime numbers?

REPLY [8 votes]: Arturo's answer is great, I would just like to add some perspectives (at least in the commutative case) that can possibly shed some more light on the subject. To begin with, it is obvious that if $x$ is an element of a (commutative) ring $R$ and $a$ and $b$ are multiples of $x$ then certainly $a + b$ is a multiple of $x$, and likewise if $a$ is a multiple of $x$ and $b$ is any element then $ab$ is a multiple of $x$. This means that the set of multiples of $x$ is closed under addition and multiplication by any element of the ring. This precisely means that this set is an Ideal, namely, the principal ideal generated by $x$. Now, we can see the defining properties of an ideal as an axiomatization of the notion of "a set of multiples of something" without reference to that mysterious "something". In PIDs, every such ideal is indeed a set of multiples of an element of the ring, but in more general rings, there are non principal ideals  which informally correspond to multiples of an "ideal element" which is just a metaphor and not a real element of the ring. One adventage of this concept comes for the observation that ideal elements have "GCD". if $a$ and $b$ are elements of a PID (for example $\mathbb{Z}$) then the ideal generated by $a$ and $b$ is precisely the principle ideal generated by their greatest common divisor. But in more general rings, this may not be a principal ideal at all (equivalently, $a$ and $b$ don't have a greatest common divisor), and in this situation we can still think of it as the ideal generated by the "ideal number" corresponding to  what should be the GCD of $a$ and $b$.
OK, so what are prime ideals then? naturally those are the "ideal number" analogies of prime numbers. In some important rings for number theory (algebraic number rings or dedekind domain is general) you don't have unique factorization for numbers but you do have unique factorization for "iedal numbers" into "prime ideal numbers".
I would like to add that even though I belive this is the historical intuition and a good one to have in mind, it is the "arithmetically minded" one. the concept of ideals and prime ideals is fundamental in general commutative rings which are very different from those you encounter in classic algebraic number theory and in particular, higher dimensional ones (whatever that means). It is important to observe that there is another and quite different intuition behind the idea of ideals. for a ring of functions from some geometric object to a field, the set of functions vanishing on a specific point, is an ideal (check this!). the notion of prime ideals can also be reached through the search for a suitable reconstruction of the "points" of the geometric object on which we would like to think the elements of our abstract ring act as functions. The ultimate result of this philosophy and the unification of it with the arithmetic one is the modern approach to algebraic-geometry via the notion of schemes.