Exercise 3, § 2

Let A be a ring and A' a ring containing A and integral over A. For every ideal $a$ of A, show that the radical of $a$ is the intersection of A and the radical of $aA'$. Show that the latter is the set of elements $x \in A'$ satisfying an equation of integral dependence whose coefficients (other than the dominant coefficient) belong to $a$ (note that, if $y, z$ are two elements of A' satisfying such an equation of integral dependence, so does $y - z$).