Simple $R$-module

Question

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true:

1) $N$ has a finite number of submodules.

2) $\operatorname{Hom}_R(N,N)$ is a division ring.

3) $\operatorname{Hom}_R(N,N)$ is isomorphic to a product of four division rings.

4) $J(R)N=0$.

Answer 1

There is a maximal ideal m of R s.t. $M=R/m$.
$N$ has finite number submodules and as Eric Towers said inverse of $φ$ is not $1-1$ so $Hom_R(N,N)$ is not a division ring. $Hom_R(N,N)≅Hom_R(R/m⨁R/m,R/m⨁R/m)$. $J(R)N=0$ because $J(R)⊆m$ so $J(R). R/m=0$