Relative Tor Functors for Level Modules with Respect to a Semidualizing Bimodule

Let R and S be rings and (S) C (R) a semidualizing bimodule. We investigate the relative Tor functors defined via C-level resolutions, and these functors are exactly the relative Tor functors defined by Salimi, Sather-Wagstaff, Tavasoli and Yassemi provided that S = R is a commutative Noetherian ring. Vanishing of these functors characterizes the finiteness of -projective dimension. Applications go in two directions. The first is to characterize when every S-module has a monic (or epic) C-level precover (or preenvelope). The second is to give some criteria for the isomorphism between the bifunctors.