Given a monomial ideal
$$I = \langle x^{\alpha} \mid \alpha \in A \subset \Bbb Z^n_{\geq 0}\rangle,$$
I want to know if the rest of the polynomial division of $f \in \mathbb{K}[x_1,...,x_n]$ by the monomial basis of $I$ is unique. In particular, is a monomial basis (eventually a reduced one) a Groebner basis? By going through examples, it seems that the rest of such a division is unique, but I would like to have a proof. Can somebody provide an explanation, a reference or a solution proposal? Thanks.
A (possible infinite) collection of monomials generating an ideal $I$ is a Gröbner basis (commutative, non-commutative, or in more general settings for other algebraic structures). To see it, notice that if two monomials overlap, then they reduce to zero no matter which one you choose to rewrite, so the conditions of the Diamond Lemma are satisfied.