# Maximal Ideal, Prime Ideal

• Sep 28th 2008, 10:56 AM
roporte
Maximal Ideal, Prime Ideal
Let K a field, prove:

i) $\langle X,Y\rangle$ is a maximal ideal in $K[X,Y]$

ii) $\langle Y\rangle$ is a prime ideal in $K[X,Y]$ that not is maximal

Thanks!!!
• Sep 28th 2008, 02:39 PM
ThePerfectHacker
Quote:

Originally Posted by roporte
i) $\langle X,Y\rangle$ is a maximal ideal in $K[X,Y]$

I see two ways of doing that. The first way is to let $\left< X,Y \right> \subseteq I\subseteq K[X,Y]$. Now $\left< X,Y\right>$ consists of all expressions involving $X,Y$. Thus for $\left< X,Y \right> \subset I$ it is necessary that $a\in I - \{ 0\}$ for some $a\in K$. But any ideal which contains a unit must be the improper ideal, so $I = K[X,Y]$. Thus, $\left$.

Another way is to form $K[X,Y]/\left< X,Y\right>$ and realize that this is the consists of only constants i.e. it is isomorphic to $K$ which is a field. Therefore, $\left< X,Y\right>$ must be a maximal ideal.

Quote:

ii) $\langle Y\rangle$ is a prime ideal in $K[X,Y]$ that not is maximal
Form $K[X,Y]/\left< Y\right>$ and realize this is modding out the $Y$'s and so it is isomorphic to $K[X]$. This is an integral domain. It follows that $\left< Y\right>$ must be a prime ideal.