Thread: Maximal Ideal, Prime Ideal

Let K a field, prove:

i) is a maximal ideal in

ii) is a prime ideal in that not is maximal

Thanks!!!

Originally Posted by roporte

i) is a maximal ideal in

I see two ways of doing that. The first way is to let . Now consists of all expressions involving . Thus for it is necessary that for some . But any ideal which contains a unit must be the improper ideal, so . Thus, .

Another way is to form and realize that this is the consists of only constants i.e. it is isomorphic to which is a field. Therefore, must be a maximal ideal.

ii) is a prime ideal in that not is maximal

Form and realize this is modding out the 's and so it is isomorphic to . This is an integral domain. It follows that must be a prime ideal.