Residual and fixed modules
The article presents some sufficient conditions for the commutativity of transvections with elements of linear groups over division ring in the language of residual and fixed submodules. The residual and fixed submodules of the element $\sigma $ of the linear group are defined as the image and nucle...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2020-06-01
|
| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/6 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849415405296156672 |
|---|---|
| author | Yu. V. Petechuk V. M. Petechuk |
| author_facet | Yu. V. Petechuk V. M. Petechuk |
| author_sort | Yu. V. Petechuk |
| collection | DOAJ |
| description | The article presents some sufficient conditions for the commutativity of transvections with elements of linear groups over division ring in the language of residual and fixed submodules. The residual and fixed submodules of the element $\sigma $ of the linear group are defined as the image and nucleus of the element $\sigma -1$ and are denoted by $R(\sigma)$ and $P(\sigma)$ respectively. It is proved that transvection ${\sigma }_1$ over an arbitrary body commutes with an element ${\sigma }_2$ for which $\mathop{\rm dim}R({\sigma }_2)=\mathop{\rm dim}R({\sigma }_2)\cap P({\sigma }_2)+l$, $l\le 1$, if and only if the inclusion system $R({\sigma }_1)\subseteq P({\sigma }_2)$, $R({\sigma }_2)\subseteq P({\sigma }_1)$. It is shown that for $l>1$ this statement is not always true. |
| format | Article |
| id | doaj-art-28f88e8b3a984c81928f395d02ed4cb0 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2020-06-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-28f88e8b3a984c81928f395d02ed4cb02025-08-20T03:33:32ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202020-06-0153211912410.30970/ms.53.2.119-1246Residual and fixed modulesYu. V. Petechuk0V. M. Petechuk1Transcarpathian Institute of Postgraduate Pedagogical Education, Uzhgorod, UkraineTranscarpathian Institute of Postgraduate Pedagogical Education, Uzhgorod, UkraineThe article presents some sufficient conditions for the commutativity of transvections with elements of linear groups over division ring in the language of residual and fixed submodules. The residual and fixed submodules of the element $\sigma $ of the linear group are defined as the image and nucleus of the element $\sigma -1$ and are denoted by $R(\sigma)$ and $P(\sigma)$ respectively. It is proved that transvection ${\sigma }_1$ over an arbitrary body commutes with an element ${\sigma }_2$ for which $\mathop{\rm dim}R({\sigma }_2)=\mathop{\rm dim}R({\sigma }_2)\cap P({\sigma }_2)+l$, $l\le 1$, if and only if the inclusion system $R({\sigma }_1)\subseteq P({\sigma }_2)$, $R({\sigma }_2)\subseteq P({\sigma }_1)$. It is shown that for $l>1$ this statement is not always true.http://matstud.org.ua/ojs/index.php/matstud/article/view/6linear groups over rings and division rings, residual and fixed modules, transvections, unipotent elements, conditions of commutativity. |
| spellingShingle | Yu. V. Petechuk V. M. Petechuk Residual and fixed modules Математичні Студії linear groups over rings and division rings, residual and fixed modules, transvections, unipotent elements, conditions of commutativity. |
| title | Residual and fixed modules |
| title_full | Residual and fixed modules |
| title_fullStr | Residual and fixed modules |
| title_full_unstemmed | Residual and fixed modules |
| title_short | Residual and fixed modules |
| title_sort | residual and fixed modules |
| topic | linear groups over rings and division rings, residual and fixed modules, transvections, unipotent elements, conditions of commutativity. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/6 |
| work_keys_str_mv | AT yuvpetechuk residualandfixedmodules AT vmpetechuk residualandfixedmodules |