Gorenstein projective, injective and flat modules over trivial ring extensions

We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let [Formula: see text] be a trivial extension of a ring [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] such that [Formula: see text] is a generalized compatible [Formula: see text]-[Formula: see text]-bimodule and [Formula: see text] is a generalized compatible [Formula: see text]-[Formula: see text]-bimodule. We prove that [Formula: see text] is a Gorenstein projective left [Formula: see text]-module if and only if the sequence [Formula: see text] is exact and coker([Formula: see text]) is a Gorenstein projective left [Formula: see text]-module. Analogously, we explicitly characterize Gorenstein injective and flat modules over trivial ring extensions. As an application, we describe Gorenstein projective, injective and flat modules over Morita context rings with zero bimodule homomorphisms.