A homework question from my algebra class asks:
Show that in a local ring $R$ with maximal ideal $M$, every element outside $M$ is a unit.
My argument is that since $M$ is maximal $R /M $ is a field and so for any $ x \in R \backslash M $, $ x + M $ has a multiplicative inverse, which implies $ x $ is a unit.
I don't see where we need the fact that $R$ is a local ring.
Best Answer
I hope you know the following theorem:
This is a standard consequence of Zorn's lemma. In particular this implies that the set of units of $R$ coincides with the complement of the union of maximal ideals of $R$.