Pseudo-Localization Of Multiplicatively Closed Set And Characterizations Of PVMRs

PVMRs have received a good deal of attention continuously in a number ofliterature in recent decade. In this thesis, PVMRs are characterized by utiliz-ing pseudo-localization principle and w-operations. In chapter 1, the concept ofpseudo-localization R[S] and S-cancellation are introduced. Then, we obtain someproperties of pseudo-localization R[S]. Moreover, the pseudo-local to global prin-ciple is received by using pseudo-localization R[S] and w-operations. In chapter2, Manis valuation theory is presented. Pseudo-valuation ring R is defined, whichhave a unique regular maximal ideal P and (R,P) is a valuation pair. On theone hand, by using pseudo-localization principle and w-operations, it is shownthat R is a PVMR if and only if (R[Q],[Q]R[Q]) is a Manis valuation ring for anymaximal w-ideal Q of R; which is equivalent to that (R[P],[P]R[P]) is a pseudo-valuation ring for any regular maximal w-ideal P of R. On the other hand, byusing traditional ideal theory methods, we proved that R is a PVMR if and onlyif every finite type ideal of R is w-projective(w-flat). If R is a weak DW ring,then R is a PVMR if and only if R is a Pru|¨fer ring.