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?

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 $R$?