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 $M/{\rm Soc} M$ has finite uniform dimension then $M$ decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics of the Czech Academy of Science
2023-07-01
|
| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/148/2/mb148_2_5.pdf |
| _version_ | 1797837582435876864 |
|---|---|
| author | Figen Takil Mutlu Adnan Tercan Ramazan Yaşar |
| author_facet | Figen Takil Mutlu Adnan Tercan Ramazan Yaşar |
| author_sort | Figen Takil Mutlu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-09T15:28:07Z |
| format | Article |
| id | doaj.art-92f8b1588f5a49ebb3ac11a872d627d9 |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-04-09T15:28:07Z |
| publishDate | 2023-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-92f8b1588f5a49ebb3ac11a872d627d92023-04-28T12:19:40ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362023-07-01148221122210.21136/MB.2022.0100-21MB.2022.0100-21Eventually semisimple weak $FI$-extending modulesFigen Takil MutluAdnan TercanRamazan YaşarIn 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.http://mb.math.cas.cz/full/148/2/mb148_2_5.pdf $cs$-module weak $cs$-module uniform dimension ascending chain on essential submodules $c_{11}$-module $fi$-extending weak $fi$-extending |
| spellingShingle | Figen Takil Mutlu Adnan Tercan Ramazan Yaşar Eventually semisimple weak $FI$-extending modules Mathematica Bohemica $cs$-module weak $cs$-module uniform dimension ascending chain on essential submodules $c_{11}$-module $fi$-extending weak $fi$-extending |
| title | Eventually semisimple weak $FI$-extending modules |
| title_full | Eventually semisimple weak $FI$-extending modules |
| title_fullStr | Eventually semisimple weak $FI$-extending modules |
| title_full_unstemmed | Eventually semisimple weak $FI$-extending modules |
| title_short | Eventually semisimple weak $FI$-extending modules |
| title_sort | eventually semisimple weak fi extending modules |
| topic | $cs$-module weak $cs$-module uniform dimension ascending chain on essential submodules $c_{11}$-module $fi$-extending weak $fi$-extending |
| url | http://mb.math.cas.cz/full/148/2/mb148_2_5.pdf |
| work_keys_str_mv | AT figentakilmutlu eventuallysemisimpleweakfiextendingmodules AT adnantercan eventuallysemisimpleweakfiextendingmodules AT ramazanyasar eventuallysemisimpleweakfiextendingmodules |