Gorenstein injective and strongly cotorsion modules.
\textit{E. E. Enochs} and \textit{O. M. G. Jenda} introduced the notion of Gorenstein injective modules as a generalization of injective modules [in Math. Z. 220, No. 4, 611-633 (1995; Zbl 0845.16005)]. They called a module \(M\in\text{Mod\,}R\) to be Gorenstein injective if there exists an exact sequence \(\cdots\to I_1\to I_0\to I^0\to I^1\to\cdots\) in \(\text{Mod\,}R\) with all terms injective, such that \(M=\text{Im}(I_0\to I^0)\) and the sequence is still exact after applying the functor \(\Hom_R(I,-)\) for any injective left \(R\)-module \(I\). A ring \(R\) is called a Gorenstein ring if \(R\) is both left and right Noetherian and if \(R\) has finite self-injective dimension on both left and the right. A left \(R\)-module \(M\) is called a cotorsion module if \(\text{Ext}^1(F,M)=0\) for any flat left \(R\)-module \(F\) and \(M\) is called strongly-cotorsion if \(\text{Ext}^1(X, M)=0\) for any left \(R\)-module \(X\) with finite flat dimension. The paper under review shows that if \(R\) is a Gorenstein ring with the injective envelope of \(_RR\) flat, then a left \(R\)-module \(M\) is Gorenstein injective if and only if \(M\) is strongly cotorsion. As a consequence, it is shown that for an Auslander Gorenstein ring \(R\), a left \(R\)-module \(M\) is Gorenstein injective if and only if \(M\) is strongly cotorsion.
scientific article; zbMATH DE number 6221323
scientific article