Does there exist such a module?

Question

Does there exist a nonzero $R$-module $M$ so that $M \cong M\oplus M$, where here $R$ is any ring with unity?

I'm not sure if this is true; I don't think it's true but I can't prove that. Clearly $M$ must be infinite for this to be true.

Answer 1

One example is a vector space of infinite dimension $\kappa$, since the direct sum will have a basis of cardinality $\kappa + \kappa = \kappa$.