# Principal Ideals and associates

I'm wondering something about principal ideals which I'm using to prove something.
K-field, let f,h be non-constant in K[X]. If f and h are associates, does it follow (f) = (h) ?
I tried to just prove it myself but I'm not sure if it's correct.
f, h associates means f=ch for some unit c. Then (f) = {fg : g in K[X]} = {hcg : g in K[X]}. Now I want to put " = (h) " but I'm not sure if that's correct. I think it is, because g runs over all of K[X], and cK[X] = K[X], so {hcg : g in K[X]} = {hq : q in K[X]}. So is my statement true?

lavinia
Gold Member
(f) = {fg : g in K[X]} = {hcg : g in K[X]} shows that any element in the ideal generated by f is in the ideal generated by g. Since c is a unit the converse is also true.

(f) = {fg : g in K[X]} = {hcg : g in K[X]} shows that any element in the ideal generated by f is in the ideal generated by g. Since c is a unit the converse is also true.
Did you mean h?

Also, is the equality {hcg : g in K[X]} = {hq : q in K[X]} incorrect?

lavinia