An ideal in is a subring such that for any and , the product is in .
If is an ideal such that , then is called proper.
A maximal ideal is a proper ideal such that whenever is an ideal with , in fact . In other words, is maximal with respect to the partial order given on ideals of by set inclusion.
For , define Note that is an ideal in .
Answer the following yes-no questions:
- Is is a maximal ideal of for all ?
- Does there exist a maximal ideal of that is not of the form for some ?