A property of $J$-semisimple rings I'd like a little help on how to begin this problem.  
Show that a PID $R$ is Jacobson-semisimple $\Leftrightarrow$ $R$ is a field or $R$ contains infinitely many nonassociate irreducible elements.  
Thanks.

If $R$ is a PID and has infinitely many nonassociated irreducible elements, then given any nonunit $x\in R$ you can find an irreducible element that does not divide $x$; can you find a maximal ideal that does not contain $x$? If so, you will have proven that $x$ is not in the Jacobson radical of $R$. The case where $R$ is a field is pretty easy as well.
Conversely, suppose $R$ is a PID that is not a field, but contains only finitely many nonassociated primes; can you exhibit an element that will necessarily lie in every maximal ideal of $R$? 
