Some remarks on equivalences between categories of modules

Let R be a ring with identity, Mod-R the category of all right R-modules, P∈Mod-R, A=End(PR), and T=(--⊗AP):Mod-A→Mod-R and H=HomR(P,--): Mod-R→Mod-A the adjoint functors. PR is self-small if H(P(X))≈A(X) canonically for any set X. PR is w-Σ-quasi-projective if H is exact on the exact sequences [#] of the form 0→L→P(X)→M→0 with L∈Gen(PR), the full subcategory of Mod-R of all P-generated right R-modules and X any set. Let Q be a cogenerator of Mod-R and K=H(Q). Then the author shows, among other things, the following: PR is self-small and w-Σ-quasi-projective if and only if (T,H) induces an equivalence of the following conditions: (1) (T,H) induces an equivalence between Cogen(KA), the full subcategory of Mod-A of all K-cogenerated right A-modules, and Pres(PR), the full subcategory of Mod-R of those modules M in the above exact sequences [#]. This result was also obtained by A. del Rio Mateos and M. Saorín Castaño [Tsukuba J. Math. 15 (1991), no. 1, 1--18] independently. The main theorem states the equivalence of the following conditions: (1) (T,H) induces an equivalence between Cogen(KA) and Gen(PR). (2) Gen(PR)=Pres(PR), PR is self-small and w-Σ-quasi-projective. (3) PR is self-small and for any set X, any M↪P(X), M∈Gen(PR) if and only if Ext1R(P,M)↪Ext1R(P,P(X)) canonically. PR is a ∗-module if it satisfies one of the above equivalent conditions. The author relates ∗-modules with tilting modules studied by D. Happel and C. M. Ringel [in Representations of algebra (Puebla, 1980), 125--144, Lecture Notes in Math., 903, Springer, Berlin, 1981; MR0654707 (83g:16055)] by the "Ext-projective'' property for P as follows: For a ∗-module PR, Ext1R(P,M)=0 for any M∈Gen(PR) if and only if Gen(PR) is a torsion class. In case PR generates any injective module, PR is a ∗-module if and only if PR is self-small and Gen(PR)={M∈Mod-R| Ext1R(P,M)=0}.