Modules with chain conditions up to isomorphism

We study modules with chain conditions up to isomorphism, in the following sense. We say that a right module M is isoartinian if, for every descending chain M≥M1≥M2≥… of submodules of M, there exists an index n≥1 such that Mn is isomorphic to Mi for every i≥n. A ring R is right isoartinian if RR is an isoartinian module. Similarly we define isonoetherian and isosimple modules and rings. We determine a number of properties of such modules and rings, giving several examples. For instance, we prove that a ring R is a right isoartinian semiprime right noetherian ring if and only if R is a finite direct product of matrix rings over principal right ideal domains.