Atiyah & Macdonald - Commutative Algebra, p.65.
Let $F$ be a field and $\Omega$ be an algebraically closed field.
Let $\Sigma$ be the collection of all $(A,f)$ such that $A$ is a subring of $F$ and $f:A\rightarrow \Omega$ is a ring homomorphism. We partially order $\Sigma$ as follows:
$$ (A,f)\leq (A',f') \text{ iff } A\subset A', f'\upharpoonright A=f$$
The text asserts that there exists a maximal element of $\Sigma$.
However, shouldn't we need more condition on $\Omega$ so that it has a maximal element? There can be no such pair $(A,f)$ because $\Sigma$ can possibly be the empty set. (This is because we are defining ring homomorphisms to send $1$ to $1$)
What would be a mild condition on $\Omega$ to have $\Sigma$ nonempty?
The set $\Sigma$ is nonempty iff either $F$ has characteristic $0$ or $F$ has positive characteristic and $\Omega$ has the same characteristic. Indeed, if $F$ has characteristic $0$ then $\mathbb{Z}$ is is a subring, and there exists a homomorphism $\mathbb{Z}\to\Omega$. If $F$ has positive characteristic $p$, then $p=0$ in any subring of $F$ so no subring can map to $\Omega$ unless $\Omega$ also has characteristic $p$. And if $\Omega$ does have characteristic $p$, then $\mathbb{F}_p$ is a subring of $F$ which has a homomorphism to $\Omega$.