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...
Gespeichert in:
| Datum: | 2018 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
|
| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-963 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2018-05-13T10:43:19Z |
| collection |
OJS |
| language |
English |
| topic |
torsion (torsion theory) Morita context torsion free module accessible module equivalence 16S90 16D90 |
| 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 |
| topic_facet |
torsion (torsion theory) Morita context torsion free module accessible module equivalence 16S90 16D90 |
| format |
Article |
| author |
Kashu, A. I. |
| author_facet |
Kashu, A. I. |
| author_sort |
Kashu, A. I. |
| title |
On equivalence of some subcategories of modules in Morita contexts |
| title_short |
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_sort |
on equivalence of some subcategories of modules in morita contexts |
| 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 \({\cal J}_{I}=\{_{R}M\ |\ IM=M\}.\) |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963 |
| work_keys_str_mv |
AT kashuai onequivalenceofsomesubcategoriesofmodulesinmoritacontexts |
| first_indexed |
2025-07-17T10:31:47Z |
| last_indexed |
2025-07-17T10:31:47Z |
| _version_ |
1837890142802542592 |