Modules satisfying double chain condition on non-small submodules

In this article, we study modules that satisfy the double infinite chain condition on non-small submodules, denoted by ns-DICC. Using this concept we extend some of the basic results of DICC modules to ns-DICC modules. We show that if an R-module M satisfies the double infinite chain condition on non-small submodules, then M has non-small Krull dimension. Moreover, we observe that an R-module M is ns-DICC if and only if for any non-small summand A of M, either A satisfies the descending chain condition on non-small submodules, or M/A is Noetherian.