**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)?