Units of Local Rings
Hi, I'm trying to prove that the units of a local ring R with maximal ideal M are exactly the elements of R-M. I checked Google and there are proofs available, so I don't need a full proof, I just have 2 questions about the proof:
1. It requires an assumption that any proper ideal I of R is contained in M. If I were asked to prove this assumption, would I have to appeal to Zorn's lemma and show the hypotheses for the lemma hold in this case?
2. During the proof, this:
ab = 1 + x, some x in R
follows from this:
(a + M)(b + M) = 1 + M.
I see how we have ab + M = 1 + M, but then how do we get from that, that ab = 1 + x, some x\in R?