Gaussian polynomials and content ideals

We prove that every regular Gaussian polynomial over a locally Noetherian ring has invertible content ideal. We do this by first proving that Gaussian polynomials over an approximately Gorenstein local ring have principal content ideal. We also show over locally Noetherian rings that a regular polynomial $f$ of degree $n$ is Gaussian if $c(fg) = c(f)c(g)$ for polynomials $g$ of degree at most $n$.