Stability of Gorenstein X-flat modules

Abstract

In this paper we introduce the notion of Gorenstein X-flat R-module and study a kind of stability of the class of Gorenstein X-flat R-modules. A ring R is called right GXF-closed if the class of all Gorenstein X-flat right R-modules is closed under extensions. We give an answer for the following natural question in the setting of a right GXF-closed ring R: Given an exact sequence of Gorenstein X-flat right R-modules G = · · ·→G₁ → G₀ → G⁰ → G¹ →· · · such that the complex G ⊗_RH is exact for each Gorenstein X-injective left R-module H, is themodule M:= im(G₀ → G⁰) a Gorenstein X-flat R-module?