  The connection between a univariate polynomial having locally principal
content and the content function acting like a homomorphism (the so-called
Gaussian property) has been explored by many authors. In this work, we extend
several such results to the contexts of multivariate polynomials, power series
over a Noetherian ring, and base change of affine $K$-algebras by separable
algebraically closed field extensions. We do so by using the framework of the
Ohm-Rush content function. The correspondence is particularly strong in cases
where the base ring is approximately Gorenstein or the element of the target
ring is regular.
