Let $P$ be a prime ideal. Suppose that $R/P$ has no nonzero nilpotent elments.
Show that $R/P$ is integral domain.
What I did :
WTS : $(a+P)(b+P)=ab+P=0+P$ implies $a+P=0+P$ or $b+P=0+P$.
but it means that $ab \in P$ implies $a \in P$ or $ b \in P$. The nilpotent condition means that $a \n...
@BillDubuque Although I agree that poor users misuse "hint" style, it is very biased to talk about "obscure hints", or of a policy that "encourag[es] bad answers".
Maybe the answer does not have to be that detailed - you can always refer to Lam's book for the auxiliary result. (Or if you can find it somewhere else.)
Google returns some reasonably looking hits, like this blog.
@PedroTamaroff The point is not to argue that specific viewpoint, but to give an example of the many viewpoints, in particular one on the opposite end of the spectrum, (and held by a very talented teacher).
Let $P$ be a prime ideal. Suppose that $R/P$ has no nonzero nilpotent elments.
Show that $R/P$ is integral domain.
What I did :
WTS : $(a+P)(b+P)=ab+P=0+P$ implies $a+P=0+P$ or $b+P=0+P$.
but it means that $ab \in P$ implies $a \in P$ or $ b \in P$. The nilpotent condition means that $a \n...
