Let $I$ be a proper nontrivial ideal of a ring $R$.
Show that $I/I^{2}$ is a superfluous submodule of $R/I^{2}$.
What I've tried:
If $J/I^{2}$ is such that $J/I^{2} + I/I^{2} = R/I^{2}$, then $\forall\ r \in R$, $\exists\ i \in I$, $\exists\ j \in J$ such that $r + I^{2} = i + j + I^{2}$, that is, $r - (i + j) \in I^{2}$, but I cannot prove that $r - j \in I^{2}$ because $i$ doesn't have to be in $I^2$.
Do you have any clue?
Every nilpotent ideal is going to be superfluous because it’s contained in the Jacobson radical, which is always a superfluous submodule of the ring (on the right and on the left.)
Another way to see it is that if $A^2=\{0\}$ and $A+B=R$, then $R=R^2=(A+B)^2\subseteq B$, so $B=R$.
Either of these is enough to see $I/I^2$ is superfluous in $R/I^2$.