PRUFER ${\upsilon}$-MULTIPLICATION DOMAINS IN WHICH EACH t-IDEAL IS DIVISORIAL

We give several characterizations of a TV-PVMD and we show that the localization R[X;S]$_{N_{\upsilon}}$ of a semigroup ring R[X;S] is a TV-PVMD if and only if R is a TV-PVMD where $N_{\upsilon}\;=\;\{f\;{\in}\;R[X]{\mid}(A_f)_{\upsilon} = R\}$ and S is a torsion free cancellative semigroup with zero.