Let $R=\mathbb{R}[X_1,\dots,X_n]$, and

$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$

be partially ordered by inclusions. Is there an algorithm or criterion that can determine if a given ideal $I$ is maximal in $\mathfrak{I}_2$. If an ideal is not maximal, then the algorithm should provide a list of all ideals in $\mathfrak{I}_2$ that contain $I$.

This question arose from my [previous question] about ideals 1 for which Pace Nielsen provided very nice examples: $$I_1:= \langle x_1 x_2, x_1(x_2^4+x_1^2)-x_2^3(x_1 x_2) \rangle = \langle x_1x_2,x_1^3 \rangle \subsetneq I_2:= \langle x_1x_2,x_2^4+x_1^2 \rangle . $$

In this case $I_1$ is not maximal but $I_2$ might be. For example, is $gcd(I)=1$ sufficient?

I am even more interested in the same question for **subrings**, let
$$\mathfrak{S}_d=\{ \text{subrings of $R$ for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ subrings generated by $d$ monomials}.\}$$

Given a subring $S$ how can we determine if it is maximal or what are larger subrings with the same number of minimal generators.

I placed the bounty for an answer dealing with the question about subrings.