Gorenstein flat-cotorsion modules over tensor rings

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize the Gorenstein flat-cotorsion modules over $T_R(M)$, showing that a $T_R(M)$-module $(X,u)$ is Gorenstein flat-cotorsion if and only if $u$ is monomorphic and $\text{Coker}u$ is a Gorenstein flat-cotorsion $R$-module. As applications, we describe the Gorenstein flat-cotorsion modules over some trivial extension rings and Morita context rings.