Gorenstein flat-cotorsion 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]. Sufficient and necessary conditions are established for Gorenstein flat-cotorsion modules over [Formula: see text]. More precisely, it is proved that under some assumptions [Formula: see text] is a Gorenstein flat-cotorsion left [Formula: see text]-module if and only if Coker[Formula: see text] is a Gorenstein flat-cotorsion left [Formula: see text]-module and the sequence [Formula: see text] is exact. As applications, it is characterized Gorenstein flat-cotorsion modules over Morita rings with zero bimodule homomorphisms.