  In this paper we study when the dual of a $t$-ideal in a $PVMD$ is a ring?
and we treat the question when it coincides with its endomorphism ring. We also
study particular classes of overrings of PVMDs. Specially, we investigate the
Nagata transform and the endomorphism ring of ideals in PVMDs in an attempt to
establish analogues for well-known results on overrings of Pr\"ufer domains.
