Content of a polynomial

The displayed equation is not true in general. For example, let $A=k[t,u]$, $k$ a field and $t, u$ indeterminates. Let $f=t+ux$ and $g=t-ux$. Then $c(fg) = c(t^2 - u^2 x^2) = (t^2, u^2)$, but $c(f)c(g) = (t,u)^2 = (t^2, tu, u^2)$, a strictly larger ideal of $A$.

A ring $A$ for which the displayed equation does hold for all $f,g \in A[x]$ is sometimes called a Gaussian ring. It is closely related to the condition of being a Prüfer domain, and indeed holds whenever $A$ is Prüfer.