On equivalence of some subcategories of modules in Morita contexts
A Morita context \((R,\,_{R}\!V_{S},\,_{S}\!W_{R},S)\) defines the isomorphism \({\cal L}_{0}(R) \cong {\cal L}_{0}(S)\) of lattices of torsions \(r\geq r_{\scriptscriptstyle I}\) of \(R\)-\(Mod\) and torsions \(s\geq r_{\scriptscriptstyle J}\) of \(S\)-\(Mod\), where \(I\) and \(J\) are the trace i...
Збережено в:
| Дата: | 2018 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543368859877376 |
|---|---|
| author | Kashu, A. I. |
| author_facet | Kashu, A. I. |
| author_sort | Kashu, A. I. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-05-13T10:43:19Z |
| description | A Morita context \((R,\,_{R}\!V_{S},\,_{S}\!W_{R},S)\) defines the isomorphism \({\cal L}_{0}(R) \cong {\cal L}_{0}(S)\) of lattices of torsions \(r\geq r_{\scriptscriptstyle I}\) of \(R\)-\(Mod\) and torsions \(s\geq r_{\scriptscriptstyle J}\) of \(S\)-\(Mod\), where \(I\) and \(J\) are the trace ideals of the given context. For every pair \((r,s)\) of corresponding torsions the modifications of functors \(T^{W}=W\otimes _{R}\)- and \(T^{V}=V\otimes _{S}\)- are considered:\[R\textrm{-}Mod\supseteq \mathcal{P}(r)\begin{array}{c}\begin{array}{c}\underrightarrow{\quad\bar{T}^W=(1/s)\cdot T^W\quad}\\\overleftarrow{\quad\bar{T}^V=(1/r)\cdot T^{V}\quad}\end{array}\end{array}\mathcal{P}(s)\subseteq S\textrm{-}Mod,\]where \({\cal P}(r)\) and \({\cal P}(s)\) are the classes of torsion free modules. It is proved that these functors define the equivalence \begin{equation*} {\cal P}(r)\cap {\cal J}_{I}\approx {\cal P}(s)\cap {\cal J}_{J}, \end{equation*} where \({\cal P}(r)=\{_{R}M\ |\ r(M)=0\}\) and \({\cal J}_{I}=\{_{R}M\ |\ IM=M\}.\) |
| first_indexed | 2026-02-08T07:58:14Z |
| format | Article |
| id | admjournalluguniveduua-article-963 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:58:14Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-9632018-05-13T10:43:19Z On equivalence of some subcategories of modules in Morita contexts Kashu, A. I. torsion (torsion theory), Morita context, torsion free module, accessible module, equivalence 16S90, 16D90 A Morita context \((R,\,_{R}\!V_{S},\,_{S}\!W_{R},S)\) defines the isomorphism \({\cal L}_{0}(R) \cong {\cal L}_{0}(S)\) of lattices of torsions \(r\geq r_{\scriptscriptstyle I}\) of \(R\)-\(Mod\) and torsions \(s\geq r_{\scriptscriptstyle J}\) of \(S\)-\(Mod\), where \(I\) and \(J\) are the trace ideals of the given context. For every pair \((r,s)\) of corresponding torsions the modifications of functors \(T^{W}=W\otimes _{R}\)- and \(T^{V}=V\otimes _{S}\)- are considered:\[R\textrm{-}Mod\supseteq \mathcal{P}(r)\begin{array}{c}\begin{array}{c}\underrightarrow{\quad\bar{T}^W=(1/s)\cdot T^W\quad}\\\overleftarrow{\quad\bar{T}^V=(1/r)\cdot T^{V}\quad}\end{array}\end{array}\mathcal{P}(s)\subseteq S\textrm{-}Mod,\]where \({\cal P}(r)\) and \({\cal P}(s)\) are the classes of torsion free modules. It is proved that these functors define the equivalence \begin{equation*} {\cal P}(r)\cap {\cal J}_{I}\approx {\cal P}(s)\cap {\cal J}_{J}, \end{equation*} where \({\cal P}(r)=\{_{R}M\ |\ r(M)=0\}\) and \({\cal J}_{I}=\{_{R}M\ |\ IM=M\}.\) Lugansk National Taras Shevchenko University 2018-05-13 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963 Algebra and Discrete Mathematics; Vol 2, No 3 (2003) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963/492 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | torsion (torsion theory) Morita context torsion free module accessible module equivalence 16S90 16D90 Kashu, A. I. On equivalence of some subcategories of modules in Morita contexts |
| title | On equivalence of some subcategories of modules in Morita contexts |
| title_full | On equivalence of some subcategories of modules in Morita contexts |
| title_fullStr | On equivalence of some subcategories of modules in Morita contexts |
| title_full_unstemmed | On equivalence of some subcategories of modules in Morita contexts |
| title_short | On equivalence of some subcategories of modules in Morita contexts |
| title_sort | on equivalence of some subcategories of modules in morita contexts |
| topic | torsion (torsion theory) Morita context torsion free module accessible module equivalence 16S90 16D90 |
| topic_facet | torsion (torsion theory) Morita context torsion free module accessible module equivalence 16S90 16D90 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963 |
| work_keys_str_mv | AT kashuai onequivalenceofsomesubcategoriesofmodulesinmoritacontexts |