Exercise 5.2 in Atiyah-Macdonald asks to show the following:
"Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an algebraically closed field $\Omega$. Show that $f$ can be extended to a homomorphism of $B$ into $\Omega.$"
My question: what are some notable counterexamples of this property, which occur when we take a ring $B$ which fails to be integral over a subring $A$?
Although I don't doubt that this property fails in general (i.e. for "most" rings), the only example I can come up with seems seriously contrived and unnatural: let $B$ be the complex numbers, let $A$ be $\mathbb{Z},$ let $\Omega$ be some algebraically closed field included in $\mathbb{C}$ (say the complex algebraic numbers) with $f$ the inclusion map.