In Atiyah and Macdonald, Chapter 5, Exercise 10, there defines the so called "going-down property" (GDP). Then in Chapter 7, Exercise 24, the hint says, the ring map $f: A\rightarrow B$ has GDP implies, for prime ideals $a\subset b$, if there is a prime ideal $b'$ of $B$ such that $b=f^{-1}(b')$, then there is a prime ideal $a'$ of $B$ such that $a=f^{-1}(a')$. Of course this is true iff $\ker (f) \subset a$. Then if $\ker (f) \notin a$, then what can we say by using GDP? My goal is to remedy the hint of Chapter 7, Exercise 24.
• Why is the going down property of $f:A \rightarrow B$ true if and only if $kerf \subset a$? – Manos Feb 13 '14 at 1:30
• Since the kernel is the inverse image of 0. I didn't say the going down property of f:A→B true if and only if kerf⊂a, I only said "for $a$"...The other direction is true by thinking about quotient ring. – Lao-tzu Feb 13 '14 at 1:30