Let be a field and let . Let be a monomial ideal in , and let be a monomial. For each , let be a greatest common divisor of and , and write .
- Prove that .
- Deduce that if is a finitely generated monomial ideal, then is a monomial ideal.
We begin with a lemma.
Lemma: (Euclid) Let be a unique factorization domain, and let such that and are relatively prime. Then . Proof: If , then certainly and . Suppose now that and . If is a unit, then certainly . Suppose is not a unit, and that is an irreducible factor of of multiplicity . Since , we have not dividing . In particular, . By induction, . Hence , and thus is a greatest common divisor of and . Now let . If is a unit, then certainly is also a greatest common divisor of and . Suppose is not a unit. We have . If is an irreducible factor of of multiplicity , then does not divide ; thus . Hence . Now suppose and ; then , and so . THus $ltex e$ is a greatest common divisor of and .
Now we show that .
Suppose . Then , and so . Say , with and the are monomials. Then . Now write as a sum of monomials; then . In particular, each term of has the form for some . Recall that is a greatest common divisor of and ; say . Then we have where and are relatively prime. Since is a unique factorization domain, by Euclid’s lemma we have . By this previous exercise, we have .
Suppose ; say . For each , say . Now , and so . Thus , and we have .