nimon's proof works only for commutative
domains because
is not necessarily infinite even if
is infinite. the answer to
xixi's question is also negative for non-commutative rings.
for example
the ring of quaternions over
has this property because every non-zero element of
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
be an infinite commutative (resp. non-commutative) ring. suppose that the number of non-unit elements of
is finite. is
necessarily a field (resp. division ring)?