[mercedesbenz]

Let $C_i$ be an infinite cyclic group in additive notation generated by $a_i$for all $i\in I$. Prove that $\sum_{i\in I}C_i$is a free abelian group having $\{v_{i}|i\in I\}$as a basis, where for all $i\in I$,
$v_{i}=\{x_j\}_{j\in i}$ where $x_j=a_i ; i=j$ and $x_j=0 ; i\neq j$

Poof: We want to show that
$\sum_{i\in I}C_i=<\{v_{i}|i\in I\}>$.
Consider $<\{v_{i}|i\in I\}>=\{n_{i_1}v_{i_1}+n_{i_2}v_{i_2}+\cdots+
n_{i_k}v_{i_k}:i\in I,n_{i_j}\in \mathbb{Z},1\leq j \leq k\}$ (It's wrong or right)
Clearly $\{v_{i}|i\in I\}>\subset \sum_{i\in I}C_i $.
then I want to show $\sum_{i\in I}C_i \subset <\{v_{i}|i\in I\}>$.
Please help show next step.....
thank you for any help.