Chain conditions on non-parallel submodules

In this paper, we investigate modules with ascending and descending chain conditions on non-parallel submodules. We call these modules np-Noetherian and np-Artinian respectively, and give structure theorems for them. It is proved that any np-Artinian module is either atomic or finitely embedded. Also, we give a sufficient condition for np-Noetherian (resp., np-Artinian) modules to be Noetherian (resp., Artinian). We study ascending (resp., descending) chain condition up to isomorphism on non-parallel submodules as npi-Noetherian (resp., npi-Artinian) modules and characterize these modules. It is shown that any npi-Noetherian module has finite type dimension. Next, we investigate some properties of semiprime right np-Artinian (resp., npi-Artinian) rings. In particular, it is proved that if $ R $ semiprime ring such that $ J(R) $ is not atomic, then $ R $ is right np-Artinian if and only if it is semisimple. Further, it is shown that if $ R $ is a semiprime right npi-Artinian ring, then either $ Z(R) $ is atomic or $ R $ is right non-singular. Finally, we investigate when np-Artinian (resp., np-Noetherian) rings and ne-Artinian (resp., ne-Noetherian) rings coincide.