I suppose we know that ...
Suppose is a commutative ring with no zero divisors.
Suppose that for every , , we have .
Prove R is a field.
I am not sure what to do... just need to show that each has an inverse (ie, each element is a unit)? Not sure how to go about it.
I think I made some progress? not sure if it is valid...
Since , then we can write each element as for some .
But since itself is in , then for some .
Since is commutative,
Does this mean that we have a unity? How do we prove that works for every element of ?
Once we can show we have a unity, I think the rest is done, because:
if , then . So each element of can be written as for some . Since , for some . Since is commutative, , so by the definition of unique inverses, each element has an inverse, so each element is a unit, so is a field.
Can anyone look over this argument and verify it/spot a problem? I'd be very grateful, thanks!