Does there exist a ring such that all modules are decomposable?
If we consider an infinite dimensional $k$-linear space $V$, then $V\cong V \oplus V $, then we can show that $R \cong R \oplus R$ as $R$-modules, where $R: = End_k(V)$ (this is an example from Rotman <An introduction to homological algebra> Example 2.36). We know $R$ is a von Neumann regular ring which does not have IBN. Maybe $R$ satisfy the condition but I can't prove it.