$fg$ primitive $\to$ $f, g$ primitive

Question

I am currently trying to prove that over $A[x]$ if $fg$ is primitive,then $f = a_0 + a_1x +\ldots + a_nx^n$ and $g = b_0 + b_1x + \ldots + b_mx^m$ are primitive (Atiyah exercise 1.3).

I defined the "coefficient"-ideals $K:=(a_0b_0, a_1b_0+a_0b_1, \ldots, a_nb_m)$, $I:=(a_0, \ldots, a_n)$ and $J:=(b_0, \ldots, b_m)$.
Obviously, $I\cdot J \supset K$.

Therefore, if $fg$ is primitive, $K = (1)$ and thus $I\cdot J = (1)$.

Now the only step would be to show that this implies $I = (1)$ and $(J) = 1$. However, I am stuck here, how do I go about proving this?

Answer 1

Obviously, $I\supset K$ and $J\supset K$.