Somewhat related to this question, I'm looking for sources that give a proof of $$S / I \cong S/\mathrm{LT}(I),$$ where $I \lhd S = k[x_1, ..., x_n]$ for $k$ a field, and the isomorphism is an isomorphism as vector spaces.
This seems to be true because $S$ is Noetherian and hence $I$ is finitely generated, meaning that for $I = \langle f_1, ..., f_r \rangle$ where WLOG $\{f_i\}$ are monic, then any monomial which is divisible by some $\mathrm{LT}(f_i)$ is replaced by a sum of lower degree monomials (here we can take $\{f_i\}$ to be a Groebner basis), so for the sake of counting basis elements in a vector space isomorphism, quotienting out by the leading term achieves the same effect.
More generally, I'm looking for sources and/or proofs about this statement when $k$ is not a field (in particular, when $k = \mathbb{Z}$), so that there'd be a $k$-module isomorphism, though I'm unsure if this even holds due to modified description of Groebner bases over PIDs.