Geodesic
Eventually semisimple weak $FI$-extending modules
Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 211-222
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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 $M=M_1\oplus M_2$ 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.
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 $M=M_1\oplus M_2$ 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.
DOI : 10.21136/MB.2022.0100-21
Classification : 16D50, 16D80
Keywords: $CS$-module; weak $CS$-module; uniform dimension; ascending chain on essential submodules; $C_{11}$-module; $FI$-extending; weak $FI$-extending 
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Takıl Mutlu, Figen; Tercan, Adnan; Yaşar, Ramazan. Eventually semisimple weak $FI$-extending modules. Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 211-222. doi: 10.21136/MB.2022.0100-21

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