Can every element of the completion $\widehat{A}$ of a local ring $A$ be written as an element of $A$ with a unit of $\widehat{A}$?

Question

Let $A$ be a noetherian local ring, $\widehat{A}$ its completion. Can every element $f\in\widehat{A}$ be written as a product $f'u$ with $f'\in A$ and $u\in\widehat{A}^\times$?

Answer 1

No,try with $A=k[x,y]_{(x,y)}, \widehat{A} = k[[x,y]]$ and take $f = x-g(y)$ with $g(y) \in y k[[y]]$ non-algebraic over $k[y]$.

Then $\widehat{A}/(f)\cong k[[g(y),y]]$ and there is no non-zero $F(x,y)\in k[x,y]_{(x,y)}$ such that $F(g(y),y)=0$ ie. $f\widehat{A}\cap k[x,y]_{(x,y)}=\{0\}$.