Duality In ∏-coherent Rings And ∏-coherence Of Strongly Almost Excellent Extensions

Fl-coherent rings are called strong coherent rings in [Joh] and [Jon]. The earliest well-known characterization of Il-coherence was given in [Ca]. In [Ca] Camillo proved the following equivalences: (i) R is right It-coherent; (ii) R is a left *-ring; (iii) for each n^-1, right annihilators of subsets of Rn is finitely generated. The class of such rings has been studied by many authors in literature such as [W], [CTHW] and [CY]. In particular, [CY] gives more comprehensive characterization of Fl-coherent rings. In this paper, we make further investigation of H-coherent rings.In chapter 2, we study the duality in It-coherent rings. We introduce WQF-rings, GIF-rings and SGQF-rings via the reflexiviry of finitely generated modules. The reflexivity of W" -modules over Fl-coherent rings is investigated. We also discuss the existence of a type of solutions of the homological equation A=Ext"R (X, R) over an SGQF-ringR. The followings are the main results we obtain.On the characterization for WQF-rings, we haveTheorem 2.1.1 Let R be a Il-coherent ring. Then the following statements are equivalent:(i) R is a WQF-ring;(ii) both RR and RR are infective cogenerators; (iii) R is a cogenerator ring; (iv) RRR defines a Morita duality; (v) RR and RR are injective and R has the double annihilator property;(vi) all cyclic left and right R-modules are reflexive. Theorem 2.1.2 Every WQF-ring is a QF-ring.?Since every QF-ring is a WQF-ring, the results of Theorem 2.1.2 show clearly the coincidence of WQF-ring and QF-ring. Considering strict inclusion{Noetherian rings}c{ri-coherent rings}, our Theorem 2.1.2 strictly generalizes the classical results on QF-rings.Xue Weimin [XI] illustrated by an example that a cogenerator ring is not necessarily Noetherian. Our Theorem 2.1.2 states that if R is a cogenerator ring that is Fl-coherent then ./? must be Noetherian and Artinian.About the characterization for GIF-rings, we haveTheorem 2.1.3 Let R be a U-coherent ring. Then the following statements are equivalent:( i) Risa GIF-ring;( ii) gR and RR are FGT-injective modules;(Hi) every projectile left R.-module and every projectile right R-module areFGT-injective modules; (iv) every infective left R-module and every infective right R-module are FGT-flatmodules; ( v) FP-id (RR)