Let be a commutative ring and let be a proper ideal. Let denote the set of maximal ideals in containing ; is nonempty by Proposition 11 in the text. Define . By convention, . Also, is called the Jacobson radical of .
- Prove that is an ideal of containing .
- Prove that .
- Let be an integer and let . Describe in terms of the prime factorization of .
We begin with a lemma.
Lemma: If are relatively prime, then . Proof: we can write by Bezout’s identity for some integers and , so that .
- By Proposition 11 in the text, is not empty. Thus by this previous exercise, is an ideal of . Since for all , we have .
- Let , and let . Now for some ; let be minimal with this property. If , then ; since is prime, we have or , a contradiction. Thus , and we have . Thus for all , and hence .
- Write the prime factorization of as . Recall that the maximal ideals of are precisely those of the form for prime . Now precisely when divides ; thus consists precisely of the ideals for the primes dividing . Thus . Using the lemma (with induction) and this previous exercise, .