Motivation behind the Krull Dimension of a ring I have been studying the Krull Dimensions of rings and have a couple of questions:
Why is the concept of Krull dimensions important within algebraic geometry? 
I have also been proving the result that the Krull dimension of the polynomial ring $K[X_1,...,X_n]$ is equal to $n$ when $K$ is a field. Why is this result in particularly important? In other words, why do we care about this result?
If anyone has any information/references on this, I would find it very interesting. 
Note: Not references on the proof as I have been able to complete this!
Thanks, 
Johan
 A: In short, Krull dimension is an important concept because it is an intrinsically algebraic definition of dimension which precisely captures our geometric intuition for what the (topological) dimension of a variety/scheme should be.
For example, we associate $\mathbb{A}^n_{\mathbb{C}} = \mathrm{Spec} \,\mathbb{C}[x_1 \ldots x_n]$ with the topological space $\mathbb{C}^n$, although the two are not homeomorphic. The fact that the Krull dimension of $\mathbb{C}[x_1 \ldots x_n]$ is $n$ captures this.
This then allows us to formalize the notions of dimension and co-dimension in such a way that they can be generalized to arbitrary rings (such as $\mathbb Z$) where no geometric intuition is available, and to study them from a geometric point of view. We can take geometric proofs that rely on the concept of dimension, translate them into algebraic language, and then prove the same statements in much greater generality. 
A very simple example of this principle is the fact that points in $\mathrm{Spec}\, A$ correspond to prime ideals of $A$, but our geometric intuition for ''points'' is really restricted to the maximal ideals of $A$. The inclusions of these points correspond to maps $A \to A/\mathfrak{m}$, which has Krull dimension 0. Hope this helps. 
A: The concept of Krull dimension is not only important in algebraic geometry, but in much more generality. In particular, for algebraic number theory and commutative algebra it is very useful and natural. The rings of integers $\mathcal{O}_K$ for a number field $K$ naturally arise in number theory as Noetherian, integrally closed rings with Krull dimension $1$. The most basic example is for $K=\mathbb{Q}$, namely $\dim (\mathbb{Z})=1$. Then
$$
\dim \mathcal{O}_K[x_1,\ldots ,x_n]=n+1.
$$
More generally, dim $A[x_1,\ldots ,x_n]=\dim (A)+n$ for Noetherian rings $A$.
