Proving Less Than 4 Monomials Can't Generate Ideal J

[Metric_Space]

Homework Statement

Show that the ideal J=(a^2, abc, ac^2, c^3) cannot be generated by less than 4 monomials.

Homework Equations

None

The Attempt at a Solution

I was thinking of computer a Groebner basis for this (which is what I ended up doing) However, I'm not sure how I can show there it cannot be generated by less than 4 monomials.

I do know that this ideal is indeed also a Groebner basis, but not sure where to go from here.

 

[mighty2000]

Thank you for your question. To show that the ideal J=(a^2, abc, ac^2, c^3) cannot be generated by less than 4 monomials, we can use the fact that the dimension of a polynomial ring is equal to the number of variables in the ring. In this case, the polynomial ring has 3 variables (a, b, c), so the dimension is 3.

Now, let's consider the generators of the ideal J. Each generator contains at least 2 variables, so the maximum number of monomials that can be generated from these generators is 2. For example, a^2 can only generate a^2 and a^2b; abc can only generate abc and abc^2, and so on.

Since the dimension of the polynomial ring is 3 and the maximum number of monomials that can be generated from the generators is 2, it is impossible to generate the ideal J with less than 4 monomials. Therefore, it is not possible to generate the ideal J with less than 4 monomials.

I hope this helps to clarify the solution. Please let me know if you have any further questions.