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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?

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  • $\begingroup$ Is it your intention that $-1=1$? Or do you want to add that as a (possibly different) unit? $\endgroup$
    – 2'5 9'2
    Sep 23, 2011 at 7:17

1 Answer 1

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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.

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