EVENTUALLY SEMISIMPLE WEAK FI-EXTENDING MODULES

In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C-3 properties and M/Soc M 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, C-3 properties and ascending (or descending) chain condition on essential submodules then M = M-1 circle plus M-2 for some semisimple submodule M-1 and Noetherian (or Artinian, respectively) submodule M-2. 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 C-11, or FI-extending, respectively) module.