TITLE: Is any subring of a field which contains the identity is itself a subfield
QUESTION [7 upvotes]: Prove that or give counterexample(s):

Any subring of a field which contains the identity is a subfield

REPLY [9 votes]: If $R$ is a finite subring of a field $F$, then it is a subfield. This follows from the fact that a finite submonoid of a group is a subgroup. Let $r\in R$, $r\ne 0$ and consider the map $f\colon R\to R$ given by $f(x)=rx$. This is injective because $R$ is a domain, hence also surjective because $R$ is finite; since $1\in R$, you find $x\in R$ with $rx=1$.
Actually this is a general fact: every finite domain is a field, the argument is exactly the same.
For infinite $R$ this isn't true, as shown in other answers.