# What is the profinite completion of a free abelian group of infinite rank?

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 $$\Bbb{Q}^{\times}\cong {\{\pm1\}}\times \bigoplus_{n\ge1} \Bbb{Z}$$