Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$-adic completion and define $\widehat{R}=R\otimes_A \widehat{A}$.
Do we have a 1:1-correspondence between the isomorphism classes of simple left $R$-modules and the isomorphism classes of simple left $\widehat{R}$-modules, induced by the map $S \rightarrow S\otimes_A \widehat{A}$?
Since $R$ is module-finite over $A$, we have that $\widehat{\mathfrak{m}}\widehat{R}\\subset rad(\widehat{R})$. So given a simple $\widehat{R}$-module $S$, then $\widehat{\mathfrak{m}}$ annihilates $S$, that is $S$ is in fact a simple $R\otimes_A\widehat{A}/\widehat{\mathfrak{m}}$-module. But $\widehat{A}/\widehat{\mathfrak{m}}=A/\mathfrak{m}$. So $S$ is a simple $R/\mathfrak{m}R$-module. But since $\mathfrak{m}\subset rad(R)$, this implies that $S$ is a simple $R$-module.
Now given a simple $R$-module $M$, then this is a finite length $A$-module, so the map $M\rightarrow M\otimes_A\widehat{A}$ is an isomorphism. But is this also an isomorphism of $\widehat{R}$-modules?
Or is this eventually not true? Is there literature about the behaviour of $R$-modules under completion of the base ring?
I ask this, because it is often easier to find simple modules if the base ring $A$ is complete, e.g. if $A/\mathfrak{m}$ is algebraically closed and $R$ is an Azumaya algebra, then $\widehat{R}\cong M_s(\widehat{A})$, and the simple modules for $\widehat{R}$ are easy to see, but for $R$?
