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Math Help - Maximal Ideal, Prime Ideal

  1. #1
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    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!!!
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  2. #2
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    Quote Originally Posted by roporte View Post
    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<X,Y\right>.

    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.

    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.
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