Let and be commutative rings and let be a ring homomorphism.
- Prove that if is a prime ideal then is a prime ideal. Apply this to the special case when to deduce that if is prime in then is prime in .
- Prove that if is a maximal ideal and if is surjective, then is maximal. Give an example to show that this need not be the case if is not maximal.
- By §7.3 #24, is an ideal of . Now suppose . Then , so that since is prime, either or . Thus either or . Hence is a prime ideal of .
Note that if is the inclusion map, then .
- Let be maximal, and note that is an ideal. Note that since is surjective. Let denote the natural projection. Since is surjective, is a surjective ring homomorphism and is a field. Moreover, . Now is a field, and thus has only the trivial ideals. Using the Lattice Isomorphism Theorem and since , we have . Since is a field, is maximal in .
Now let be a maximal ideal and consider the inclusion map . Then is not maximal in .