I am studying for a qualifying exam, and have been working on this problem:
Let $D$ be a PID and let $P$ be a prime ideal of the polynomial ring $D[x]$. Suppose that $P$ contains a non-zero constant polynomial. Prove that $P$ can be generated by two elements.
I saw this, but I'm having trouble generalizing it for rings that aren't $\mathbb{Z}$. I know that $D[x]$ is also an integral domain (not a PID) so I don't get that $D[x]/P$ is a field, and I'm trying to understand the case where $P=(c,f(x),g(x))$ for some constant $c$ and two nonconstant polynomials $f(x),g(x)$. Does it work to replace $f$ and $g$ with their gcd? If so, can someone please explain why?
Thanks for your help.
If $P$ contains a non-zero element of $D$, then it contains a prime element of $D$ (why?), say $p$. Then $P/pD[x]$ is a prime ideal of $D[x]/pD[x]$. But $D[x]/pD[x]$ is isomorphic to $(D/pD)[x]$ (why?) and $D/pD$ is a field, so $D[x]/pD[x]$ is a PID. This shows that $P/pD[x]$ is principal. It follows that $P$ can be generated by two elements.