Direct sum of indecomposable modules Prove or disprove: There exist indecomposable modules $V,U,W$ such that $\displaystyle{ V \oplus U \cong V \oplus W }$ and $\displaystyle { U \not \cong W}$.

I think that this is true, but I have no idea how to construct such indecomposable modules.

Any ideas?
Modules over what?  If we are looking at $R$-modules for $R$ a field, for example, then only one-dimensional spaces are indecomposable and there are no such examples.

Over $\mathbb{Z}$ there should be some interesting examples (hint: Hilbert's Hotel cannot be modeled directly, but there are some infinite groups that might behave similarly...).  In general, this is a hard and interesting problem without assumptions on $R$.  This paper, for example, says some interesting things in the case where $R$ has dimension 1.
HINT: Take the quotient of both sides with respect to V.