Given a known ring $R$ and an ideal $I\subseteq R$, I would like to be able to solve an "ideal polynomial equation" of the form $$\sum_i I_i^{\alpha_i}=I,$$ i.e to find ideals $I_i$ so that the above equation holds. I would want to have a list of all possible solutions.
An example of something I would want to solve is the ideal equation $$(IJ)^5+I^3J^2+I^2J^3+I=R$$ where $R$ is a ring of the form $R=k[x_1, \cdots, x_m]/K$ where $K$ is some known finitely general ideal of $R$.
Could this be coded in Macaulay2?