Eventually semisimple weak $FI$-extending modules

Abstract

In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module.