Hello, I am stuck with these questions
1. Show that is a maximal ideal in
2. Let be a commutative ring with unity. Prove that every prime ideal of is also a maximal ideal of .
Thanks in advance for your help.
1) Assume some ideal of strictly contains you ideal, i.e. there is an element in . Can you prove that the unity belongs to .
2) That's wrong. Take an integral domain which is not a field. Then is a prime ideal which is not maximal. A less extreme case: is a prime ideal but not maximal (why?).
Were there other hypotheses or was it maximal implies prime (which is true)?