There is a nice geometric example by Zariski to show why going down theorem (http://en.wikipedia.org/wiki/Going_up_and_going_down) fails when the bottom ring is not normal.

Take a cylinder over a node and call its coordinate ring $A$. Let $B$ be the normalization of $A$. Let $P$ be a singular point of Spec$(A)$ and $Q_1,Q_2$ be points of Spec$(B)$ lying above $P$. Take an irreducible curve $C$ in Spec$(B)$ passing through $Q_1$ but avoiding $Q_2$. Then there no sequence in $A$ corresponding to the inclusion of prime ideals $I(C) \subset I(Q_1)$.