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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 ?

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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.

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

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