Originally Posted by
NonCommAlg nimon's proof works only for commutative domains because $\displaystyle rU$ is not necessarily infinite even if $\displaystyle U$ is infinite. the answer to xixi's question is also negative for non-commutative rings.
for example $\displaystyle \mathbb{H},$ the ring of quaternions over $\displaystyle \mathbb{R},$ has this property because every non-zero element of $\displaystyle \mathbb{H}$ is a unit. in general, every (infinite) division ring has the property because every
non-zero element of a division ring is a unit. here's a less trivial version of xixi's problem:
let $\displaystyle R$ be an infinite commutative (resp. non-commutative) ring. suppose that the number of non-unit elements of $\displaystyle R$ is finite. is $\displaystyle R$ necessarily a field (resp. division ring)?