By definition, profinite completion of a group $G$ is $\widehat{G}=\varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$. Let $M=\bigoplus_{n\ge1} \Bbb{Z}$ be a free abelian group of countably infinite rank.
$1$. What is $\widehat{M}$?
My guess is $\widehat{M}=\prod_{n\ge1}\Bbb{\widehat{Z}}$. Am I right? How can I prove?
$2$. More generally, what is $\widehat{\oplus_{n\ge1}{ C_n}}$ where $C_n$ is cyclic group? Is it ${\prod_{n\ge1}{\widehat{ C_n}}}$?
Similarly what is pro-$p$-completions?
My questioins are originated from the profinite completion of $\Bbb{Q}^{\times}$, the multiplicative group of the rational number field.
It is known that $\Bbb{Q}^{\times}\cong {\{\pm1\}}\times \bigoplus_{n\ge1} \Bbb{Z} $