By Robert R. Colby

This publication presents a unified method of a lot of the theories of equivalence and duality among different types of modules that has transpired during the last forty five years. extra lately, many authors (including the authors of this ebook) have investigated relationships among different types of modules over a couple of earrings which are brought on by way of either covariant and contravariant representable functors, particularly, by means of tilting and cotilting theories. gathering and unifying the elemental result of those investigations with leading edge and simply comprehensible proofs, the authors' supply an relief to additional study in this valuable subject in summary algebra.

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4. Conversely, assume that (i), (ii), and (iii) hold. 4, (i) and (ii) imply that Gen(VR ) ⊆ V ⊥ . So assume M ∈ V ⊥ . Since Gen(VR ) ⊆ V ⊥ , TrV (M) ∈ V ⊥ . Hence, the exact sequence 0 → β Tr(M) → M → M/ Tr(M) → 0, where β is the inclusion, induces an exact sequence Hom(V,β) 0 → Hom R (V, TrV (M)) −→ Hom R (V, M) → Hom R (V, M/TrV (M)) → 0 where Hom(V, β) is epic by definition of trace, and so Hom R (V, M/ Tr(M)) = 0. But we also have the exact sequence 0 = Ext1R (V, M) → Ext1R (V, M/ TrV (M)) → 0; thus M/ Tr(M) = 0 by (iii), and we conclude that M ∈ Gen(VR ).

Let S VR be a tilting bimodule. Then R is a right hereditary ring if and only if (1) the induced torsion theory (S, E) in Mod-S splits and (2) proj . dim . N ≤ 1 for all N ∈ E. Moreover, if these conditions hold, then (3) inj . dim . N ≤ 1 for all N ∈ S. Proof. (⇒). Assume that R is right hereditary. 3. 7. 1, yields an exact sequence 0 = Ext2S (N , P) −→ Ext2S (N , X ) −→ Ext3S (N , K ) = 0 that establishes (2). 1 that if M ∈ F = Ker H in Mod-R, then inj . (H M) ≤ inj . dim . M. Thus (3) follows since S = H F.

4) This follows at once from (2) and (3). 2. Assuming the notation of the Tilting Theorem, we make the following observations: 1. The torsion submodules of M ∈ Mod-R and N ∈ Mod-S are τT (M) = TrV (M) = ν M (T H M)) ∼ = T H M and τS (N ) = Ann N ( S V ) = Ker η N ∼ = H T M. 2. Since S is closed under epimorphic images and direct sums, F is closed under submodules and direct products, and T : S F : H is an equivalence, letting R VS = H ( R R R ), we see that VS is a ∗-module with End(VS ) ∼ = T H (R) ∼ = R/τT (R) such that T ∼ = Hom S (V , ) on S and H ∼ = ( ⊗ R V ) on F.