Let $R=\mathbb{Z}_4$ and consider the right $R$-module $E_R=\mathbb{Z}_4\oplus\mathbb{Z}_4$ and the submodule $M=\{(\bar{0},\bar{0}),(\bar{2},\bar{2})\}$. Identify two distinct injective hulls of $M$ in $E$.
I have no idea, how to find these modules. I showed that $E_R$ is injective, but this seems to be too big. So I am searching for two injective submodules of $E_R$ which contain $M$ and $M$ is a large/essential submodule in them.
I found $Q_1=M\cup\{(\bar{1},\bar{3}),(\bar{3},\bar{1})\}$ and $Q_2=M\cup\{(\bar{1},\bar{1}),(\bar{3},\bar{3})\}$ and it is easy to check that these are submodules of $E$. As they are both isomorphic to $\mathbb{Z}_4$, they are injective modules (It is easy to see that $\mathbb{Z}_4$ is injective using Baer's criterion). To see that $M$ is a large submodule in $Q_1$ assume that there is a submodule $X$ of $Q_1$ such that $M\cap X=0$. If $X$ is non zero, then it must contain $(\bar{1},\bar{3})$ or $(\bar{3},\bar{1})$ and also twice the element, which is $(\bar{2},\bar{2})$ which is a contradiction, so $X=0$, so $M$ is large in $Q_1$ and also in $Q_2$.