Let be a field and let . Let and be monomial ideals in , where and are monomials. Prove that and .
First we show that .
Let , where and , and all but finitely many of the and are zero. Then . Let . Collecting the terms in and , respectively, we can write ; certainly then .
Now we show that .
Let ; say where and . Then . Let ; say , where all but finitely many are zero. Certainly then .