Let be a field and let . Prove that every monomial ideal in is finitely generated.
Let denote the class of all monomial ideals in which are not finitely generated by monomials. Note that is partially ordered by inclusion. Suppose by way of contradiction that is nonempty. Let be a linearly ordered collection of ideals in ; by this previous exercise, is a monomial ideal. Suppose now that is finitely generated; say by . Now each is contained in some ; since is finite, we may choose the largest such . Then , so that is in fact finitely generated- a contradiction. Thus is not finitely generated, and so is in . Thus every chain in has an upper bound in . By Zorn’s lemma, contains a maximal element .
Now by this previous exercise, we know that the prime ideals of are all finitely generated. In particular, is not prime. Thus, there exist polynomials such that but . By this previous exercise, we know that some monomial terms of and of are not in , but that (since is a monomial term of ) .
Since , we have the proper containment . By this previous exercise, is a monomial ideal. Since is maximal in the class of non-finitely generated monomial ideals, must be finitely generated- say by . Note that each of these is in either or ; relabeling if necessary, say that and . In particular, note that since otherwise . Let . Then we have . In particular, .
We claim that . To see the direction, let and . Certainly then , and so . Thus . Next, let . Then ; thus we have where and . Rearranging, we have , so that . Thus , and so . Thus we have .
Next, we note that . Then , and so . Certainly , and this containment is proper since . By this previous exercise, is a monomial ideal, and by the maximality of it is finitely generated. Then by this previous exercise, is finitely generated, a contradiction.
And so our assumption that is nonempty was faulty; in fact, every monomial ideal in is finitely generated by monomials.