Generalized Loose Edge Factorization Theorems

We extend a factorization theorem by Gwo\'zdziewicz and Hejmej from the ring of formal power series to any complete regular local ring $R$. More precisely, let $f \in R$ and assume that its Newton polyhedron has a loose edge such that the initial formal of $f$ along the latter is a product of two coprime polynomials, where one of them is not divided by any variable. Then this provides a factorization of $f$ in $R$. As a consequence we obtain a factorization theorem for Weierstra{\ss} polynomials with coefficients in $R$, which generalizes an earlier result by Rond and the author.