Let $(A,m)$ be a commutative noetherian local ring such that $m$ is principal, say $m=(t)$. Let $(\hat A,\hat m)$ be its $m$-adic completion. Let $A\subset B\subset\hat A$ be any intermediate subring such that $n=tB$ is a maximal ideal of $B$.

The question is: Is it true that the localisation $B_n$ is contained in $\hat A$?

Does this follows simply by considering the isomorphims $\hat A\simeq A[[X]]/(X-t)\simeq A[[t]]$? This would imply that an element $g\in B\setminus n$ is a unit in $\hat A$. Am I right?

Thanks in advance!