Equivalences represented by faithful non-tilting $*$-modules

Let V be an R-module, S=End(RV) and SV∗=HomR(V,Q), where RQ is a fixed cogenerator. Then V is called a ∗-module [C. Menini and A. Orsatti, Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)] if the functors HomR(V,−) and V⊗S− give an equivalence between the category of all R-modules generated by RV and the category of all S-modules cogenerated by SV∗. It is known that the class of ∗-modules contains faithful quasi-progenerators and tilting modules, and this paper deals with the question whether this containment is proper. The authors prove that if RP is a faithful quasi-progenerator with endomorphism ring A, AT is a tilting module with endomorphism ring S, and both RP and AT are not generators, then RV=RP⊗AT is a faithful ∗-module with endomorphism ring S which is neither a quasi-progenerator nor a tilting module. Moreover, if RP is projective, then RV is a quasi-tilting and a partially tilting module. Note that the above assumptions can be actually satisfied. If A is a ring with a non-projective tilting module AT, PA=A(N)A and R=End(PA), then RV=RP⊗AT is a faithful ∗-module which is neither a quasi-progenerator nor a tilting module, and it is a quasi-tilting and a partially tilting module.