The queston was inspiried by the somewhat silly notion of whether one can "solve" equations of modules, with the sum of two $R$-modules being $\oplus$ and multiplication being $\otimes$.
When I asked our lecturer he mentioned that the right setting for the question might be algebraic K-theory, where $K_0(R)$ is something like the free module of isomorphism classes of finitely generated $R$ modules with direct sum serving as the group operation. Using tensor products of modules as the multiplication, this gives us a ring (This is what I recall, but I know nothing about algebraic K-theory besides what I just wrote, so it might be horribly wrong).
One can see that the zero module satisfies the condition and vector spaces of dimension $2$ and of infinite dimension over $k$ also satisfies the condtion. The question thus is not very specific: A "complete" classification seems very diffcult, but maybe other classes of modules satisfy the aforementoned condition.
Lastly, if anyone could shed some more light on the algebraic K-theory perspective that would certainly be much appreciated (if there is any light to be shed in this direction)!