Let K a field, prove:
i) $\displaystyle \langle X,Y\rangle$ is a maximal ideal in $\displaystyle K[X,Y]$
ii) $\displaystyle \langle Y\rangle$ is a prime ideal in $\displaystyle K[X,Y]$ that not is maximal
Thanks!!!
I see two ways of doing that. The first way is to let $\displaystyle \left< X,Y \right> \subseteq I\subseteq K[X,Y]$. Now $\displaystyle \left< X,Y\right>$ consists of all expressions involving $\displaystyle X,Y$. Thus for $\displaystyle \left< X,Y \right> \subset I$ it is necessary that $\displaystyle a\in I - \{ 0\}$ for some $\displaystyle a\in K$. But any ideal which contains a unit must be the improper ideal, so $\displaystyle I = K[X,Y]$. Thus, $\displaystyle \left<X,Y\right>$.
Another way is to form $\displaystyle K[X,Y]/\left< X,Y\right>$ and realize that this is the consists of only constants i.e. it is isomorphic to $\displaystyle K$ which is a field. Therefore, $\displaystyle \left< X,Y\right>$ must be a maximal ideal.
Form $\displaystyle K[X,Y]/\left< Y\right>$ and realize this is modding out the $\displaystyle Y$'s and so it is isomorphic to $\displaystyle K[X]$. This is an integral domain. It follows that $\displaystyle \left< Y\right>$ must be a prime ideal.ii) $\displaystyle \langle Y\rangle$ is a prime ideal in $\displaystyle K[X,Y]$ that not is maximal