$I$ nil ideal and $S/I$ is nil in $R/I$ then $S+I$ is nil. Disclaimer: R is a unital ring, not necessarily commutative.
Let $I$ be a nil ideal and $S/I \subseteq R/I$ be a nil subset (every element is nilpotent) of $R/I$ where $S$ is a subset of $R$.
Show that $S+I$ is a nil subset of $R$.
Attempt: Let $s\in S, i\in I$. We have to show that $i+s$ is nilpotent. I can see that some power of $s$ must lie in $I$ but I don't see how this helps.


I can see that some power of  must lie in  but I don't see how this helps.

You are like a millimeter away from the solution.
Using your observation, $(i+s)^n=(\text{things with at least one $i$})+s^n\in I$.  But $I$ is a nil ideal... see the solution?
