# Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials

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.

• I'm pretty sure that $I_2$ is not maximal, since it is contained in the 2-generated ideal $\langle x_1,x_2\rangle$. – Pace Nielsen Mar 10 '15 at 23:02
• @PaceNielsen: I always ask slightly wrong question. One could consider $\mathfrak{I}_d \setminus \{\text{ ideals generated by$d$monomials}\}$ instead. – warsaga Mar 11 '15 at 6:44
• Your intermediate question about whether $\gcd(I)=1$ is sufficient has a negative answer. For instance $\langle x_3^2,x_1+x_2\rangle$ is not generated by monomials, and has trivial gcd, but it is not maximal with respect to those properties since it is properly contained in $\langle x_3,x_1+x_2\rangle$. The condition of not being generated by monomials is somewhat strange for a number of reasons. For example, it is not preserved under automorphisms of the ring. – Pace Nielsen Mar 16 '15 at 2:58