Would this be correct to prove that two ideals are equal?

Question

Given that reduced Gröbner bases are unique for any given ideal and any monomial ordering, would it be correct to prove that two given ideals, $I_1$ and $I_2$, are equal following this process?

1. Choose a polynomial ordering.

2. Calculate the reduced Gröbner base of $I_1$ (using the selected ordering)

3. Calculate the reduced Gröbner base of $I_2$ (using the selected ordering)

If both results match, $I_1$ is equal to $I_2$.

Answer 1

Yes, and you don't even need the reduced property to prove that $I_1=I_2$. Let $G_1$ be any Gröbner basis for $I_1$. Let $G_2$ be any Gröbner basis for $I_2$. If $G_1=G_2$, then $I_1=\langle G_1\rangle=\langle G_2\rangle=I_2$. The point is that two ideals with the same generating set must be equal.

However, the power of reduced Gröbner bases comes in as follows: if $G_1$ and $G_2$ are reduced and $G_1\not=G_2$, then $I_1\not= I_2$. This is just because the reduced Gröbner basis for any ideal is unique (with respect to a given monomial order).

So if your goal is to algorithmically determine if two ideals are equal or not, then yes, you can compute reduced Gröbner bases to detect this.