Is there a classification of local rings with trivial group of units?
Out of curiosity, is there a classification of all local rings with trivial group of units?
I suppose what I'm trying to ask is, if I asked for all local rings $R$ with $R^\times=\{1\}$, what would they be?
Solution 1:
In a local ring, the non-units are precisely the maximal ideal, and in particular, are closed under addition. For any $x \in R$, we have $x + (1-x) = 1$, a unit, and therefore at least one of $x$ and $1-x$ is a unit. Since 1 is the only unit, it follows that either $x=0$ or $x=1$. Since $x$ was arbitrary, we conclude that $R$ is the field of 2 elements.