TITLE: Example of an integral domain which is not a field
QUESTION [7 upvotes]: I just proved that a commutative ring $R$ is an integral domain iff $R$ is isomorphic to a subring of a field. 

My question is why can't $R$ be a field with these conditions? Aren't we satisfying all of the field's properties?

Thanks.

REPLY [2 votes]: A yet another example are polynomials $R[x]$ over a field $R$. While (obviously) all other properties are fine, there are much less inverses than there are required for a field.
Now formal power series offer somewhat a remedy...