On lifting and extending properties on direct sums of hollow uniform modules
A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) an...
Saved in:
| Date: | 2022 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2022
|
| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1643 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) and any finite direct sum decomposition \(M = \bigoplus_{i = 1}^n M_i\), there exists a direct summand \(M_i'\) of \(M_i\) \((i = 1, 2, \ldots, n)\) such that \(M = X \oplus (\bigoplus_{i = 1}^n M_i')\). In this paper, we first give characterizations for the square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question ``Does any lifting module satisfy the finite internal exchange property?'' as an application of this result. |
|---|