therefore examine the ideal of $R_r^{S_r}$ generated by $\avg{P_1^{r+1},P_2,\dots,P_r}$.

Taken together, IBP and the EOM define the ideal $I_r = \avg{P_1,P_1^{r+1},P_2,\dots,P_r}$ and equivalence classes of operators lie in the quotient ring $$\bigslant{R_r^{S_r}}{I_r} = \bigslant{\mathbb{R}[P_1,\dots,P_r]}{\avg{P_1,P_1^{r+1},P_2,\dots,P_r}} =\bigslant{\mathbb{R}[P_1]}{\avg{P_1,P_1^{r+1}}}.$$ In the quotient ring, the EOM simply remove the generators $P_2, \dots, P_r$. For the one flavor case, momentum conservation also removes the generator $P_1$ and the quotient ring is trivial; it consists only of the identity element. The reason we do not indicate this in the above equation is that the above form is well suited for generalization to the $N$ flavor case. For the single flavor case, $R_r^{S_r}/I_r$ being trivial reflects the fact that we can use integration by parts and equations of motion to remove all operators with derivatives acting on $\ph$—any term of the form $\phi^{r-k} (\dd \phi)^k$ with $k>0$ can be written as a total derivative: $\dd( \phi^{r-k+1} (\dd \phi)^{k-1})$.

The generalization to $N$ flavors is straightforward and follows the exact steps as the one flavor case. Each field is Fourier decomposed with the $i$th flavor having Fourier coefficients $a_q^{(i)}$, $i = 1, \dots, N$. An operator composed of $\mathbf{r}$ powers of $\mathbf{\ph}$ fields and $k$ derivatives is decomposed in 