Gaussian elements of a semicontent algebra (Q1635338)

Given an associative commutative unitary ring \(R\), an \(R\)-module \(M\) and \(x\in M\), the (Ohm-Rush) \textit{content ideal} \(\text{c}(x)\) of \(x\) in \(M\), is the intersection of all right ideals \(A\) for which \(x\in AM\). Let \(S\) be an \(R\)-algebra and \(f\in S\). The following question is studied: when \(f\) satisfies the equality \(c(fg)=c(f)c(g)\) for each \(g\in S\)? When \(S=R[x]\) is the polynomial ring in one inderminate, there are already some well known results which are recalled in this paper. Other results are obtained when \(S\neq R[x]\) in this article. The authors summarize these new results in the following way: Main Results. Let \(R\) be a commutative ring, and let \(S\) be an \(R\)-algebra and \(f\in S\) as below. Then under any of the following conditions, \(c(f)\) is locally principal if and only if is \(c(fg)=c(f)c(g)\) for every \(g\in S\): I. \(R\) is Noetherian and approximately Gorenstein, and \(S = R[x_1,\;\dots,\;x_n]\) or any other affine semigroup algebra over \(R\). II. \(R\) is locally Noetherian, \(S = R[x_1,\;\dots,\;x_n]\) or any other affine semigroup algebra over \(R\), and \(f\) is regular. III. \(R\) is Noetherian and approximately Gorenstein, and \(S = R[[x_1,\;\dots,\;x_n]]\). IV. \(R\) is a finitely generated artinian Gorenstein \(K\)-algebra, where \(K\) is a field, \(L = K(y_1,\;\dots,\;y_t)\) is a purely transcendental field extension, and \(S = R \otimes_K L\), where ``content'' is with respect to the field variables \(y_j\) . V. \(R\) is a finitely generated \(K\)-algebra, where \(K\) is an algebraically closed field, \(L/K\) is any field extension, \(f\) is regular, and \(S = R \otimes_K L\), where ``content`` is with respect to a vector space basis of \(L\) over \(K\). where a Noetherian local ring (\(R, {\mathbf m}\)) is called \textit{approximately Gorenstein} if for any positive integer \(N\), there is an \({\mathbf m}\)-primary ideal \(I\) such that \(I \subseteq {\mathbf m}N\) and \(R/I\) is a Gorenstein ring; and a ring \(R\) is \textit{locally approximately Gorenstein} if \(R_{\mathbf m}\) is approximately Gorenstein for every maximal ideal \({\mathbf m}\) of \(R\).