Matlis reflexive modules on complete rings (Q1910750)

The authors and \textit{J. L. Bueso} [Commun. Algebra 22, No. 3, 969-987 (1994; Zbl 0794.13018)], using torsion theoretic techniques, developed a new construction of the algebraic completion of rings and modules in the following way: let \(R\) be a commutative ring and \(\sigma\leq\tau\) torsion theories in \(R\)-mod such that \(R\) is \(\sigma\)-noetherian. Then they have defined the \((\sigma,\tau)\)-completion of an \(R\)-module \(M\) to be \[ M^{(\sigma,\tau)}=\varprojlim\{Q_\sigma(M/N):M/N\in{\mathcal T}_\tau\}. \] The aim of the paper under review is to describe those \((\sigma,\tau)\)-complete rings and modules giving a structural result in terms of simpler modules. Generalizing the classical result of Matlis, it is shown that for a \(\sigma\)-finitely generated \(R\)-module \(M\), there exists a natural isomorphism between the \((\sigma,\tau)\)-completion \(M^{(\sigma, \tau)}\) and the double \((\sigma,\tau)\)-dual \(M^{**}\) of \(M\). It is also shown that \(R\) is \((\sigma,\tau)\)-complete if and only if every \(\sigma\)-finitely generated \(R\)-module is \((\sigma,\tau)\)-reflexive, and that, for a \((\sigma,\tau)\)-complete ring \(R\), an \(R\)-module \(M\) is \((\sigma,\tau)\)-reflexive if and only if there exists a finitely generated submodule \(N\) of \(M\) such that \(M/N\) is \(\sigma\)-artinian. This generalizes a result due to Enochs.