So I have a commutative ring with unity, and I also have an ideal I of the ring such that every element not in the ideal is a unit of R. I have already proven that I is maximal by showing that R/I is a field, but I'm really not sure how to sure that it is unique. Let's say I have any maximal ideal of R (possibly I itself). Now I want to show that it is equal to I. How would I do that? I'm quite stuck, so any hints would be appreciated.
Thanks in advance.