TITLE: Question about Maps of power series rings
QUESTION [6 upvotes]: Suppose we have an injective homomorphism of power series rings 
$$k[[x_1,...,x_m]]\to k[[y_1,...,y_n]]$$
can $x_i$ map to a unit in $k[[y_1,...,y_n]]$?
Of course in general, for injective homomorphisms of rings, non-units can become units in larger rings. For example, $\mathbb{Z}\to \mathbb{Q}$. I am also assuming that $k$ is fixed under the above embedding.

REPLY [2 votes]: The answer is no (injective or not) at least if you consider homomorphisms of $k$-algebras. Suppose you have such a map $k[[x_1,\dots, x_n]]\to k[[y_1,\dots, y_m]]$. Composing with the evaluation at $(0,\dots, 0)\in k^m$, you get a map 
$$ \phi: k[[x_1,\dots, x_n]]\to k $$ 
and you want to prove that $\phi(x_i)=0$ for all $i\le n$. Suppose for instance that $\phi(x_1)=\lambda\in k^*$. Then $\phi(x_1-\lambda)=0$. But $x_1-\lambda$ is an unit in $k[[x_1,\dots, x_n]]$ and 
$$ 0 = \phi(x_1-\lambda) \phi(1/(x_1-\lambda))=\phi(1).$$ 
Contradiction.