Guest rainpurple Posted January 3, 2005 Share Posted January 3, 2005 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~ Link to comment Share on other sites More sharing options...
matt grime Posted January 3, 2005 Share Posted January 3, 2005 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. Link to comment Share on other sites More sharing options...
Guest rainpurple Posted January 8, 2005 Share Posted January 8, 2005 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~ Link to comment Share on other sites More sharing options...
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