kernel maximal ideal If I have a homomorphism $\phi : R \rightarrow S$ between integral domains, how can I show that if the kernel is non-zero then it is a maximal ideal in R? 

This is true for a class $\:\mathfrak R\:$ of rings iff every ring in $\:\mathfrak R\:$ has dimension at most one, i.e. prime ideals are minimal or maximal. Equivalently, by factoring out by the (necessarily) prime kernel, it is true iff every domain in $\:\mathfrak R\:$ is a field. Examples of such one-dimensional classes of rings are Dedekind domains (e.g. PIDs), and Artinian rings (see this prior question).
