On v-Noetherian Rings

A Mori domain is an integral domain which satisfies the ascending chain condition on v-ideals.In this paper the study is extended to commutative rings with zero divisors.A v-Noetherian ring is defined to be a ring which satisfies the ascending chain condition on v-ideals.We prove that if R is a v-Noetherian ring and P is a prime ideal,then R[P] is a v-Noetherian ring.Moreover we also show that if each nonzero ideal is contained in at most finitely many maximal t-ideals,then R is a v-Noetherian ring if and only if for each maximal t-ideal M,R[M] is a v-Noetherian ring.