Let $\{R_i\}_{i\in I}$ a collection of $M$ modules. I know that if $\{e_1,...,e_m\}$ is a basis of $\oplus_{i=1}^nR_i$, then, $$x\in \oplus_{i=1}^nR_i\iff \exists ! a_1,...,a_m\in M: x=a_1e_1+...+a_me_m.$$
Now, I have a definition when $I$ is unspecified, i.e. $$x\in \oplus_{i\in I}R_i\iff x_i=0\text{ except for a number finite of $i$}.$$
Is there a correlation between those two definitions ?