Let $A\subset B$ be rings, and suppose that $B\setminus A$ is closed under multiplication. Show that $A$ is integrally closed in $B$. (Atiyah and MacDonald, Introduction to Commutative Algebra, Chapter 5, Exercise 7)
I tried localizing at $B\setminus A$, but this did not seem to work. Neither did a direct application of the definition of "integral dependence". Any suggestions?