The content of a Gaussian polynomial is invertible

Let $R$ be an integral domain and let $f(X)$ be a nonzero polynomial in $R[X]$. The content of $f$ is the ideal $\mathfrak c(f)$ generated by the coefficients of $f$. The polynomial $f(X)$ is called Gaussian if $\mathfrak c(fg) = \mathfrak c(f)\mathfrak c(g)$ for all $g(X) \in R[X]$. It is well known that if $\mathfrak c(f)$ is an invertible ideal, then $f$ is Gaussian. In this note we prove the converse.