Let R be a domain with quotient field K. It is proved that R is an integrally closed domain if and only if every nonzero t-ideal of R is complete, if and only if every nonzero v-ideal of R is complete. We also obtain that every prime ideal of an integrally closed domain is integrally closed, and every strongly prime ideal of a domain is integrally closed. Moreover, we introduce the notion of w-cancellation ideals and give some equivalent characterizations of PVMDs. In particular, it is proved that R is a PVMD if and only if every w-ideal of R is complete.
In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.