Let K a field, prove:
i) is a maximal ideal in
ii) is a prime ideal in that not is maximal
Thanks!!!
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.
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.ii) is a prime ideal in that not is maximal