My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any surjective endomorphism of a finitely generated module for a Noetherian ring is injective.
Let $R$ be a unital ring, not necessarily commutative. An $R$-module $M$ is Hopfian if any surjective $R$-module endomorphism $f : M \rightarrow M$ is injective.
Is there a ring $R$ and an $R$-module $M$ such that $M$ is Hopfian but $M \oplus M$ is not?
As a follow-up, in the event the answer is `yes', is there an example where $M=R$?