Integral Domains in which Every Nonzero t-Locally Principal Ideal is t-Invertible

Let D be an integral domain, D[X] be the polynomial ring over D, and w be the so-called w-operation on D. We define D to be a w-LPI domain if every nonzero w-locally principal ideal of D is w-invertible. This article presents some properties of w-LPI domains. It is shown that D is a w-LPI domain if and only if D[X] is a w-LPI domain, if and only if D[X]_Nvis an LPI domain, where N_v= {f ∈ D[X] {pipe} c(f)^-1= D}. As a corollary, it is also shown that D is a w-LPI domain if and only if each w-locally finitely generated and w-locally free submodule of Dⁿof rank n over D is of w-finite type. We show that if D = ∩_α D_α is a finite character intersection of t-linked overrings D_α and if each D_α is a w-LPI domain, then D is a w-LPI domain.