Let $I$ be a nilpotent left ideal of $R$. Prove that for each left $R$-module $M$, $IM\ll M$ Let $I$ be a nilpotent left ideal of $R$. Prove that for each left $R$-module $M$, $IM\ll M$.  ($\ll$ means superfluous)
If $IM + N = M$, then $I^2M + IN + N = M$
I know we will use this information in somewhere but I dont know how to prove.

Don’t quit now!
From where you started notice  $M=I^2M + IN + N = I^2M+N$.
Repeating that again, $M=I^3M+N$.
You see where this is headed?
