Some variants of ascending and descending chain conditions.

We introduce the idea of nei-Noetherian and nei-Artinian modules and rings. We discuss some examples of nei-Noetherian (nei-Artinian) modules and study several properties of them. We characterize these modules such that an R module M is nei-Noetherian (nei-Artinian) if and only if every non essential submodule of M is iso-Noetherian (iso-Artinian) if and only if every proper closed submodule of M is iso-Noetherian (iso-Artinian). We further characterize semiprime right semihereditary nei-Noetherian rings. Finally, we discuss some other variants of ascending and descending chain conditions on the classes of non summands. [ABSTRACT FROM AUTHOR]