An integral domain is called globalized multiplicatively pinched-Dedekind domain (GMPD domain) if every nonzero non-invertible ideal can be written as  with J invertible ideal and  distinct ideals which are maximal among the nonzero non-invertible ideals, cf. [2]. The GMPD domains with only finitely many overrings have been recently studied in [15]. In this paper, we find the exact number of quasi-local overrings of GMPD domains having only finitely many overrings. Also, we study the effect of quasi-local overrings on the properties of GMPD domains. Moreover, we consider the structure of the partially ordered set of prime ideals (ordered under inclusion) in a GMPD domain.

overring, localization, integrally closed, Prüfer domain, Dedekind domain, valuation domain, pseudo-valuation domain.