How to compute Krull dimension concretely

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly quotients), but I will restrict to two.

Compute the Krull dimension of the following rings. $R = \mathbb{C}[x, y]/(x^{100}, y^{100})$. And $R = \mathbb Z_p/(p^{10})$.

The book has two examples of how to compute the dimension. It proves that $\dim \mathbb Z[\sqrt 5] = 1$ and $\dim \mathbb C[x, y]/(y^2 - x^3 + x) = 1$. In the first case, it uses the map $\mathrm{Spec} \mathbb Z[\sqrt 5]$ and, in the second case, it uses the projection map $\mathbb C[x] \to R$.

The proofs are very long. Considering there are 30 examples in this problem, I don't think the authors intended such complicated solutions. My question is: what approach is typical in this case? I can exhibit particular examples of primes in $R$, but I don't know how to build a maximal chain.