TITLE: A tensor product of power series
QUESTION [26 upvotes]: Let $k$ be a field. I am wondering if there is an easy description of the ring
$$k[[x]] \otimes_{k[x]} k[[x]]$$
that is the tensor product of the power series ring $k[[x]]$ with itself over the ring of polynomials $k[x]$.
Any hints would be appreciated.
Edit: As suggested in the comments I am asking: is this ring isomorphic to $k[[x]]$?

REPLY [4 votes]: The rings $R:=k[[x]]\otimes_{k[x]}k[[x]]$ and $k[[x]]$ are not isomorphic. Here is a mild simplification of Georges Elencwajg's proof. 
Assume by contradiction $R\simeq k[[x]]$. Let $K$ be the fraction field of $R$, and $S$ the multiplicative system $k[x]\backslash\{0\}$. As $R$ is a maximal subring of $K$, and $x\otimes1$ is invertible in $$S^{-1}R=k((x))\otimes_{k(x)}k((x)),$$ but not in $R$, we have 
$$K=S^{-1}R=k((x))\otimes_{k(x)}k((x)).$$ 
If $a$ is in $k((x))\backslash k(x)$, then $a\otimes1-1\otimes a$ is a nonzero element of $K$ which is mapped to $0$ by the natural morphism to $k((x))$, a contradiction.