Eventually semisimple weak $FI$-extending modules

Figen Takıl Mutlu; Adnan Tercan; Ramazan Yaşar

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 2, page 211-222
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Takıl Mutlu, Figen, Tercan, Adnan, and Yaşar, Ramazan. "Eventually semisimple weak $FI$-extending modules." Mathematica Bohemica 148.2 (2023): 211-222. <http://eudml.org/doc/299550>.

@article{TakılMutlu2023,
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 $M=M_1\oplus M_2$ 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.},
author = {Takıl Mutlu, Figen, Tercan, Adnan, Yaşar, Ramazan},
journal = {Mathematica Bohemica},
language = {eng},
number = {2},
pages = {211-222},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Eventually semisimple weak $FI$-extending modules},
url = {http://eudml.org/doc/299550},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Takıl Mutlu, Figen
AU - Tercan, Adnan
AU - Yaşar, Ramazan
TI - Eventually semisimple weak $FI$-extending modules
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 2
SP - 211
EP - 222
AB - 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.
LA - eng
UR - http://eudml.org/doc/299550
ER -

References

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