Weak FI-extending Modules with ACC or DCC on Essential Submodules.

In this paper we study modules with the WFI+-extending property. We prove that if M satisfies the WFI+-extending, pseudo duo properties and M=(SocM) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the WFI+- extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M1 M2 for some semisimple submodule M1 and Noetherian (respectively, Artinian) submodule M2. Moreover, we show that if M is a WFI-extending module with pseudo duo, C2 and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results. [ABSTRACT FROM AUTHOR]