Let be a field, and let be a monomial ideal. Suppose is an inclusion-minimal generating set for . Prove that the are unique (up to associates and a rearrangement).
First, note that if is a monomial generating set of , then by this previous exercise, some finite subset of also generates . In particular, every inclusion-minimal monomial generating set of is finite.
Suppose is a second inclusion-minimal generating set for . Now both and are Gröbner bases for by this previous exercise. In particular, we may assume that no divides another, and similarly that no divides another, since otherwise or is not inclusion-minimal.
Now ; by this previous exercise, must be divisible by some . Likewise, is divisible by . Thus is divisible by , and by the minimality of , we have . In particular, and are associates.
Define a relation by if and only if and are associates. We have showed that for every , there exists such that . Moreover, if and , then divides , a violation of the inclusion-minimalness of . So in fact is well-defined – that is, a function. Similarly, is injective, and since and are finite sets, is a bijection.
Thus an inclusion-minimal monomial generating set of is unique up to associates and a rearrangement.