Some remarks on \(\Phi\)-sharp modules
The purpose of this paper is to introduce some new classes of modules which is closely related to the classes of sharp modules, pseudo-Dedekind modules and \(TV\)-modules. In this paper we introduce the concepts of \(\Phi\)-sharp modules, \(\Phi\)-pseudo-Dedekind modules and \(\Phi\)-\(TV\)-modules....
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/111 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-1112018-04-26T02:43:18Z Some remarks on \(\Phi\)-sharp modules Yousefian Darani, Ahmad Rahmatinia, Mahdi \(\Phi\)-sharp module, \(\Phi\)-pseudo-Dedekind module, \(\Phi\)-Dedekind module, \(\Phi\)-\(TV\) module Primary 16N99, 16S99; Secondary 06C05, 16N20 The purpose of this paper is to introduce some new classes of modules which is closely related to the classes of sharp modules, pseudo-Dedekind modules and \(TV\)-modules. In this paper we introduce the concepts of \(\Phi\)-sharp modules, \(\Phi\)-pseudo-Dedekind modules and \(\Phi\)-\(TV\)-modules. Let \(R\) be a commutative ring with identity and set \(\mathbb{H}=\lbrace M\mid M\) is an \(R\)-module and \(\operatorname{Nil}(M)\) is a divided prime submodule of \(M\rbrace\). For an \(R\)-module \(M\in\mathbb{H}\), set \(T=(R\setminus Z(M))\cap (R\setminus Z(R))\), \(\mathfrak{T}(M)=T^{-1}(M)\) and \(P:=(\operatorname{Nil}(M):_{R}M)\). In this case the mapping \(\Phi:\mathfrak{T}(M)\longrightarrow M_{P}\) given by \(\Phi(x/s)=x/s\) is an \(R\)-module homomorphism. The restriction of \(\Phi\) to \(M\) is also an \(R\)-module homomorphism from \(M\) in to \(M_{P}\) given by \(\Phi(m/1)=m/1\) for every \(m\in M\). An \(R\)-module \(M\in \mathbb{H}\) is called a \(\Phi\)-sharp module if for every nonnil submodules \(N,L\) of \(M\) and every nonnil ideal \(I\) of \(R\) with \(N\supseteq IL\), there exist a nonnil ideal \(I'\supseteq I\) of \(R\) and a submodule \(L'\supseteq L\) of \(M\) such that \(N=I'L'\). We prove that Many of the properties and characterizations of sharp modules may be extended to \(\Phi\)-sharp modules, but some can not. Lugansk National Taras Shevchenko University 2018-01-24 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/111 Algebra and Discrete Mathematics; Vol 24, No 2 (2017) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/111/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/111/26 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-04-26T02:43:18Z |
| collection |
OJS |
| language |
English |
| topic |
\(\Phi\)-sharp module \(\Phi\)-pseudo-Dedekind module \(\Phi\)-Dedekind module \(\Phi\)-\(TV\) module Primary 16N99 16S99; Secondary 06C05 16N20 |
| spellingShingle |
\(\Phi\)-sharp module \(\Phi\)-pseudo-Dedekind module \(\Phi\)-Dedekind module \(\Phi\)-\(TV\) module Primary 16N99 16S99; Secondary 06C05 16N20 Yousefian Darani, Ahmad Rahmatinia, Mahdi Some remarks on \(\Phi\)-sharp modules |
| topic_facet |
\(\Phi\)-sharp module \(\Phi\)-pseudo-Dedekind module \(\Phi\)-Dedekind module \(\Phi\)-\(TV\) module Primary 16N99 16S99; Secondary 06C05 16N20 |
| format |
Article |
| author |
Yousefian Darani, Ahmad Rahmatinia, Mahdi |
| author_facet |
Yousefian Darani, Ahmad Rahmatinia, Mahdi |
| author_sort |
Yousefian Darani, Ahmad |
| title |
Some remarks on \(\Phi\)-sharp modules |
| title_short |
Some remarks on \(\Phi\)-sharp modules |
| title_full |
Some remarks on \(\Phi\)-sharp modules |
| title_fullStr |
Some remarks on \(\Phi\)-sharp modules |
| title_full_unstemmed |
Some remarks on \(\Phi\)-sharp modules |
| title_sort |
some remarks on \(\phi\)-sharp modules |
| description |
The purpose of this paper is to introduce some new classes of modules which is closely related to the classes of sharp modules, pseudo-Dedekind modules and \(TV\)-modules. In this paper we introduce the concepts of \(\Phi\)-sharp modules, \(\Phi\)-pseudo-Dedekind modules and \(\Phi\)-\(TV\)-modules. Let \(R\) be a commutative ring with identity and set \(\mathbb{H}=\lbrace M\mid M\) is an \(R\)-module and \(\operatorname{Nil}(M)\) is a divided prime submodule of \(M\rbrace\). For an \(R\)-module \(M\in\mathbb{H}\), set \(T=(R\setminus Z(M))\cap (R\setminus Z(R))\), \(\mathfrak{T}(M)=T^{-1}(M)\) and \(P:=(\operatorname{Nil}(M):_{R}M)\). In this case the mapping \(\Phi:\mathfrak{T}(M)\longrightarrow M_{P}\) given by \(\Phi(x/s)=x/s\) is an \(R\)-module homomorphism. The restriction of \(\Phi\) to \(M\) is also an \(R\)-module homomorphism from \(M\) in to \(M_{P}\) given by \(\Phi(m/1)=m/1\) for every \(m\in M\). An \(R\)-module \(M\in \mathbb{H}\) is called a \(\Phi\)-sharp module if for every nonnil submodules \(N,L\) of \(M\) and every nonnil ideal \(I\) of \(R\) with \(N\supseteq IL\), there exist a nonnil ideal \(I'\supseteq I\) of \(R\) and a submodule \(L'\supseteq L\) of \(M\) such that \(N=I'L'\). We prove that Many of the properties and characterizations of sharp modules may be extended to \(\Phi\)-sharp modules, but some can not. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/111 |
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AT yousefiandaraniahmad someremarksonphisharpmodules AT rahmatiniamahdi someremarksonphisharpmodules |
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2025-12-02T15:34:09Z |
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2025-12-02T15:34:09Z |
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