Let be a field, let , and let and be monomial ideals in (not necessarily finitely generated). Prove that is also a monomial ideal.
Let be a least common multiple of and for each and . We wish to show that .
Suppose . By this previous exercise, every term in is divisible by some and also by some ; then every term is divisible by some . Thus .
Suppose . Then each term in is divisible by some ; hence each term is divisible by some (so that ) and also by some (so that ). So .
Note that an easy induction argument shows that in fact any finite intersection of monomial ideals is a monomial ideal.