Let $R$ be a commutative Noetherian ring with unity. Let us call an $R$-module $M$ to be Hopfian if every surjective endomorphism $M \to M $ is injective.
1) If $M_1$ and $M_2$ are Hopfian modules, then is $M_1 \oplus M_2$ necessarily Hopfian ?
2) If we have an exact sequence $0\to M_1 \to M\to M_2\to 0$ with $M$ Hopfian, then are $M_1$ and $M_2$ necessarily Hopfian ?
If these are not true in general, are there any additional conditions on $R$ that would make it true ?
Neither is true for $R=\mathbb{Z}$ (i.e., for Hopfian abelian groups).
For (1), Corner gave an example of two Hopfian abelian grops whose direct sum is not Hopfian in
Corner, A. L. S., Three examples on Hopficity in torsion-free Abelian groups, Acta Math. Acad. Sci. Hung. 16, 303-310 (1965). ZBL0145.03302.
(2) is easier. $\mathbb{Q}$ is a Hopfian abelian group, but has a quotient $\mathbb{Q}/\mathbb{Z}$ which is not Hopfian, since multiplication by $n$ is surjective but not injective for any $n>1$.