Let $I$ be a two sided ideal of a ring $R$ such that $I$ is maximal as a right ideal. I need to show that $R/I^n$ is a local ring, for every $n \geqslant 1$.
For $n=1$ I was able to show that the quotient $R/I$ is a division ring and so it is a local ring (because the non-invertible elements form a group).
For $n>1$ I tried to use induction, but got stuck. Am I on the right track? Do you have any suggestions? Thanks.