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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?