The question is if $J \subset I \subset R$ be ideals and we have that $\langle LT(I) \rangle = \langle LT(J) \rangle$ then $I = J$.
I would like to show that for all monomials in I which are not in $\langle LT(I) \rangle$, these monomials span ${\frac RI}$.
This basis is going to span ${\frac RJ}$ as well and it ends the proof. However, I can not prove that such set of monomials forms a basis.
I would appreciate it if someone would help me to solve this problem.