Here I asked how to prove that a ring is local iff the set of non-units is an ideal: Proving that ring is local iff set of non-units is an ideal
Now we are supposed to show in our homework that if $R$ is local (i.e., it contains exactly one maximal ideal) and $I\subsetneq R$ is an ideal, then $R/I$ is local. I found this previous post: Quotient ring local if Ring is local
Now, I read through the marked answer and showed that $\pi(M)$ is an ideal (easy exercise). [I use the notation $\pi: R\rightarrow R/I, r\mapsto r+I$ instead of $f$.]
So it remains to be shown that $\pi(M)$ is maximal. I thought of using the fact that $I$ is maximal iff $R/I$ is a field. Would I then - in this task - have to show that $(R/I)/I$ is a field in this task, or am I misunderstanding it? However, how could we show that $(R/I)/I$ is a field?
Kind regards, MathIsFun