A class of special formal triangular matrix rings

Let \(T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )\) $T=(\begin{array}{cc}R& 0\\ C& S\end{array})$ be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule. It is proven that (1) A left S-module M in Bass class is C-torsionless (resp. C-reflexive) if and only if \(\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,M)\\ M \end{array}\biggr )\) $(\begin{array}{c}{\text{Hom}}_S(C,M)\\ M\end{array})$ is a torsionless (resp. reflexive) left T-module; (2) A left S-module M in Bass class is C-Gorenstein projective if and only if \(\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\) $(\begin{array}{c}{\text{Hom}}_S(C,M)\\ M\end{array})$ is a Gorenstein projective left T-module; (3) If C is a faithfully semidualizing (S, R)-bimodule, then a left S-module M is C-n-tilting if and only if \(\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )\) $(\begin{array}{c}{\text{Hom}}_S(C,M)\\ S\oplus M\end{array})$ is an n-tilting left T-module.