Continuous homomorphisms and rings of injective dimension one (Q2890395)

Let \(S\) be an \(R\)-algebra and \(\mathfrak{a}\) be an ideal of \(S\). In the paper under review the authors define the continuous hom functor from \(R\)-Mod to \(S\)-Mod with respect to the \(\mathfrak{a}\)-adic topology on \(S\). It is shown that: \begin{itemize} \item(1) The continuous hom functor preserves injective modules if and only if the ideal-adic property and ideal-continuity property are satisfied for \(S\) and \(\mathfrak{a}\). \item (2) If \(S\) is \(\mathfrak{a}\)-finite over \(R\), then the continuous hom functor also preserves essential extensions.NEWLINENEWLINEUsing the continuous hom functor the authors characterize rings of injective dimension one using symmetry for a special class of formal power series subrings. In the noetherian case, this enables the authors to construct one dimensional local Gorenstein domains.\end{itemize}