# Is the power series ring over a polynomial ring the same as the polynomial ring over the power series ring?

Let $$R$$ be a commutative ring with $$1.$$ Then consider the power series ring over a polynomial ring as $$R[X][[Y]]$$ and the polynomial ring over a power series ring $$R[[Y]][X].$$ Are these two objects same ?

I think $$R[X][[Y]] \subset R[[Y]][X].$$ Is the converse also true ?

The inclusions $$R[X][[Y]] \subset R[[X]][Y]$$ or $$R[X][[Y]] \subset R[[Y]][X]$$ are false.

Consider the sequence $$a_k = X^k$$ in $$R[X]$$ and then $$\sum_k a_k Y^k = \sum X^k Y^k \in R[X][[Y]]$$. This formal power series is not in $$R[[Y]][X]$$ and not in $$R[[X]][Y]$$.

On the other hand $$R[[X]][Y] \subset R[Y][[X]]$$, by rearranging terms.

• Thank you for the nice answer. – user371231 Nov 8 '18 at 13:33