quotient module with dimensional 1

Question

If $V=\bigoplus V_{i}$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $\ n-1$ dimension). If the quotient space $V/W$ is also an $R$-module; thus it's of $\ 1$-dimension. I want to show that bais of the quotient module is $v_{i}+W$ no matter what $v_{i}\in V_{i}$ is. I mean even $v_{i}+w=v_{j}+W.$ Can you please provide me with a hint?