On Gaussian polynomials and content ideal (Q839704)

Here are the main results of this paper: \parindent=6mm\begin{itemize}\item[{\(\bullet\)}] Let \((A,M)\) be a local ring which is not a field such that \(M^2=0\), and let \(R=A\propto E\) be the trivial ring extension of \(A\) by a nonzero \(A\)-module \(E\). Assume that \(R\) satisfies the following property: \[ \text{Each Gaussian polynomial over \(R\) has locally principal content.}\tag{\(*\)} \] Then (1) \(ME\neq0\); (2) If \(E\) a free \(A\)-module, then rank\(_A (E)=1\) and \(M\) is a principal ideal. \item[{\(\bullet\)}] Let \((A,M)\) be a local ring which is not a field that satisfies property (\(*\)) and such that \(M^2=0\) . Then \(A\) is a Bézout ring. \item[{\(\bullet\)}] A finite product of rings satisfies property (\(*\)) iff each of its factors satisfies this property. \end{itemize}