On torsion Gorenstein injective modules (Q2716330)

The author continues to study Gorenstein rings and modules. He investigates what he calls Gorenstein injective rings, Gorenstein injective modules and injective envelopes of them. He characterizes Gorenstein injective modules. In the second part of the paper the crucial theorems assert, for a Gorenstein integral domain \(D\), a left \(D\)-module \(M\) and the Gorenstein injective envelope \(GM\) of \(M\), first that if \(D\) is injective and \(M\) is torsion then \(GM\) is also torsion, and secondly that the torsion submodule \(tGM\) of \(GM\) is Gorenstein injective. Moreover, it is proved that the property of being Gorenstein injective is preserved under direct limits.