For help: ideal of k[x^2,x^3] can be generated by two elements

[Guest rainpurple]

Let k[x^2,x^3] be the subring of k[x] generated by the field k and the elements x^2 and x^3

then how to see that every ideal of k[x^2,x^3] can be generated by two elements? (It's said that one of the generator can be chosen as the polynomial with smallest degree)

 

thx a lot~

 

Rp~

[matt grime]

Thinking out loud: let R be the ring, I an ideal of R. Let p be an element of minimal degree. Either p generates I and we are done, or there is some element in I not in IR. Obivously, no element in I\I(p) can have degree deg(p), otherwise if q were one then we could choose t in k such that p-tq is an element of deg less than deg p or deg q.

 

So, let q be some element in I\I(p) of smallest degree, then p and q must generate I: suppose r were an element not in the span of p and q, with minimal degree, then we can manufacture an element from combinations of p, q, x^2 and x^3 having the same degree and leading coefficient, call it s, and then r-s would be in I, and not in the ideal generated by p and q, and of smaller degree than r, so it is zero.

[Guest rainpurple]

Thanks a lot!

 

I think it works. And I also found a paper saying that any nonzero polynomial in K is a strong two-generator of K[x^2,x^3]

 

Rp~