Base change and power series rings

Question

I am confused about base change for power series rings. For a concrete example, take the $\mathbb{Z}_p$-algebra $R = \mathbb{Z}_p[[X]]$. Do we have $R \otimes \mathbb{F}_p = \mathbb{F}_p[[X]]$ ? Or do we need to take the completed tensor product to get $\mathbb{F}_p[[X]]$?

Answer 1

$$\mathbb{Z}_p [[X]]\otimes _\mathbb Z\mathbb{F}_p=\mathbb{Z}_p [[X]]\otimes _\mathbb Z \frac {\mathbb Z}{p\mathbb Z}=\frac {\mathbb{Z}_p [[X]]}{p\mathbb{Z}_p [[X]]}=\mathbb{F}_p [[X]]$$