Tensor Products of Flat Cotorsion Modules and Cotorsion Dimension

This paper studies the tensor product of flat cotorsion modules. Let~$R$~and $S$ be~$k$-algebras. We prove that both~$R$-module\ $M$ and~$S$-module\ $N$ are flat cotorsion modules if and only if~$M{\otimes }_{k}N$ is a flat cotorsion~$R{\otimes }_{k}S$-module. Based on this conclusion, we provide a lower bound for the global cotorsion dimension of the tensor product algebra~$R{\otimes }_{k}S$ under appropriate conditions.