Weakly Sigma-cotorsion rings

We study the class of rings $R$ for which every direct sum of injective $R$-modules is cotorsion. We call them weakly $\mathrm{\Sigma }$-cotorsion rings. The defining property might be seen as the dual of Chase’s characterization of coherence in terms of the flatness of every direct product of projective $R$-modules. More generally, we study rings over which direct sums of injective modules have finite cotorsion dimension and call them weakly $n$-$\mathrm{\Sigma }$-cotorsion rings, as well as rings over which direct sums of cotorsion modules have finite cotorsion dimension (called $n$-$\mathrm{\Sigma }$-cotorsion rings). In the process, we obtain new characterizations of $n$-perfect rings and extend previous results by Guil Asensio and Herzog, and by Šaroch and Šťovíček.