Determining certain units in a local ring. I've been stuck on this problem for a while:
Let R be a commutative ring with $1 \neq 0$. If R has a unique maximal ideal (i.e. R is local), then either $x$ or $1-x$ (or both) are units in R.

Suppose $I=(x)$ and $J=(1-x)$ are proper ideals.
Every proper ideal is contained in a maximal ideal, but there is only one, say $M$. Then $I \subset M$ and $J \subset M$ and so $x, (1-x) \in M \implies x+(1-x)=1 \in M$ which is absurd.
Then $I$ or $J$ or both are not proper, e.g. $I=R \implies \exists a \in R$ such that  $ax=1 \implies x$ is a unit.
