On iso-Mori domains

Let R be an integral domain. We say that R is an iso-Mori domain, if for every increasing sequence I1⊆I2⊆I3⊆⋯ of integral divisorial ideals of R, there exists n∈N such that Ik is isomorphic to Ik+1 for all k≥n. In this paper, we investigate the class of iso-Mori domains and establish a fundamental characterization. We prove that an integral domain R with quotient field K, is an iso-Mori domain if and only if for every increasing sequence I1⊆I2⊆I3⊆⋯ of integral divisorial ideals of R, there exists n∈N such that for each k≥n, there exists 0≠uk∈K such that Ik=ukIk+1 if and only if for every descending chain J1⊇J2⊇J3⊇⋯ of fractional divisorial ideals of R with ⋂k=1∞Jk≠(0), there exists n∈N such that Jk+1 is isomorphic to Jk for all k≥n. Moreover, for quasilocal domains R⊆S sharing the same prime ideals, we show that R is an iso-Mori domain if and only if S is an iso-Mori domain. Furthermore, when R is an integrally closed domain, we establish that R is an iso-Mori domain if and only if the polynomial ring R[X] is an iso-Mori domain. Finally, we provide an example of an integral domain that is an iso-Mori domain but not a Mori domain: the ring of Hahn series.