On Comparability Conditions and Generalized Valuation Maps

In this study, we generalize the comparability conditions, addressed in Comparability of ideals and valuation over rings [2], between certain maximal ideals and fractional ideals of D which also force D to be a quasi-local domain. Also, we introduce the notion of an almost totally ordered group and establish that: “An integral domain D is an AVD if and only if the group of divisibility of D is almost totally ordered”. Further we also establish the groups of divisibility for APVD and PAVD, by which we can easily define the corresponding maps which are basically the generalization of valuation map. Finally, by a similar approach as in [2], we translate these comparability conditions into conditions on the partial ordering on the groups of divisibility of D.