Example for $c(fg)\neq c(f)c(g)$

Question

Let $R$ be a ring and let $f(X)$ be a nonzero polynomial in $R[X]$. The content of $f$ is the ideal $c(f)$ generated by the coefficients of f.

The ring is called Gaussian if $c(fg)=c(f)c(g)$ for all $f,g \in R[X]$.

Do you know an example where $c(fg)\neq c(f)c(g)$?

Thanks a lot.

Answer 1

Take $R=\mathbb{F}_2[x,y]$. In order to see why $R[X]$ is not Gaussian, take $f(X)=g(X)=x+yX$. Then $fg=f^2=x^2+y^2X^2$ and therefore $$c(fg)=c(f^2)=\langle x^2,y^2\rangle.$$ On the other hand $$c(f)c(g)=c(f)^2=\langle x^2,y^2,xy\rangle.$$