Usually in algebraic geometry,we define the dimension of a topological space as follows:
$\dim(X)=\sup\{ m:\text{ there exists a descending chain of irreducible closed sets of length } m\}$
and in a commutative ring $R$ ,we define the Krull dimension as follows:
$\dim(R)=\sup\{m': \text{ there exists an ascending chain of closed sets of length } m'\}$
I know that there exists a correspondence between irreducible closed sets of $\mathbb A^n_K$ and the prime ideals of $K[X_1,X_2,...,X_n]$ for an algebraically closed field $K$, so via the correspondence we can get one from the other. But what motivated one of the definitions is not clear to me. Can someone help me with this?
We want to say $K[x_1,\ldots,x_n]$ for a field $K$ has dimension $n$. It has the chain of $n+1$ distinct prime ideals $$ \{0\} \subset (x_1) \subset (x_1,x_2) \subset \cdots \subset (x_1,\ldots,x_n). $$ and you can't fit any prime ideals inside any step of this chain. This suggests in a commutative ring $R$ looking at chains of distinct prime ideals $$ \mathfrak p_0 \subset \mathfrak p_1 \subset \cdots \subset \mathfrak p_n $$ and calling the largest such $n$ the dimension of $R$. That is the Krull dimension of $R$.