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1. Algebra and Number Theory

Direct limits of Gorenstein injective modules

Alina Iacob
Georgia Southern University, Statesboro, USA

Abstract:

One of the open problems in Gorenstein homological algebra is: when is the class of Gorenstein injective modules closed under arbitrary direct limits? It is known that if the class of Gorenstein injective modules, GI, is closed under direct limits, then the ring is noetherian. The open problem is whether or not the converse holds. We give equivalent characterizations of GI being closed under direct limits. More precisely, we show that the following statements are equivalent:
(1) The class of Gorenstein injective left R-modules is closed under direct limits.
(2) The ring R is left noetherian and the character module of every Gorenstein injective left R-module is Gorenstein flat.
(3) The class of Gorenstein injective modules is covering and it is closed under pure quotients.
(4) GI is closed under pure submodules.