Mathematica Bohemica, Vol. 148, No. 2, pp. 211-222, 2023
Eventually semisimple weak $FI$-extending modules
Figen Takıl Mutlu, Adnan Tercan, Ramazan Yaşar
Received July 2, 2021. Published online June 2, 2022.
Abstract: 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 $M/{\rm Soc} M$ 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 $M=M_1øplus M_2$ 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.
