Determining ring type for the formal series ring

Question

Given the following set of rings:

$$\mathcal{C} = \{\mathbb{Z}[i\sqrt{3}], \ \mathbb{Z}[X], \ \mathbb{Z}, \ \mathbb{Q} , \ \mathbb{Q}[X]\}$$

I am told to determine, $\forall S\in\mathcal{C}$ what type of ring the formal series of $S$, $(S[[X]],+,\cdot)$, is. That is, is it a field? An euclidean domain? Ideal principal domain? UFD?

Any help with this question would be apreciated!

Answer 1

All above power series rings have been studied at MSE. For example, $\mathbb{Z}[[x]]$ is a unique factorization domain, see here, and is a $2$-dimensional regular Noetherian domain, but not a PID, because $\mathbb{Z}$ is not a field (for $K[[X]]$ with a field $K$ it is a PID, see here). Hence it is not Euclidean as well.