Krull dimension of $\mathbb{R}[x,x^{-1}]$

Question

I had the following exercise:

1. Calculate Krull dimension of $\mathbb{R}[x,x^{-1}]$
2. Let $A$ be a Noetherian ring. Calculate the dimension of $A[x,x^{-1}]$ as a function of that of $A$.

I have carried out these points using the following theorems:

Theorem 1. Let $K$ be a field. Then dim $K[x_1,\dots,x_n]=n$.

So I deduced dim $\mathbb{R}[x,x^{-1}]=2$.

Theorem 2 Let $A$ be a noetherian ring. Then dim $A[x_1,\dots, x_n]=$ dim $A+n$.

So I supposed dim $A[x,x^{-1}]=$ dim $A+2$.

---

Now I have found that the results are wrong. Why couldn't I use these theorems? I don't understand.

And how was I supposed to do the exercise?

How could I calculate dim $\mathbb{R}[x,x^{-1}]$ and dim $A[x,x^{-1}]$?