Let be a non-commutative ring . Suppose that the number of non-units of is finite . Can we say that is a finite ring?
I think that we can say is a finite ring. The only way it could be infinite is if the group of units, call it , were infinite. But then for any with would be an infinite number of non-unital elements, which is a contradiction.
So I guess we must assume that there is at least one non-zero, non-unital element, otherwise an infinite ring of nothing but units with an additive identity thrown in (such as ) would destroy the argument.
In short: yes, unless is a field!
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)?