Generalization of primal superideals
Let \(R\) be a commutative super-ring with unity \(1\not=0\). A proper superideal of \(R\) is a superideal \(I\) of \(R\) such that \(I\not=R\). Let \(\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper superide...
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| Datum: | 2016 |
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Lugansk National Taras Shevchenko University
2016
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/26 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543306033397760 |
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| author | Jaber, Ameer |
| author_facet | Jaber, Ameer |
| author_sort | Jaber, Ameer |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2016-07-12T10:09:40Z |
| description | Let \(R\) be a commutative super-ring with unity \(1\not=0\). A proper superideal of \(R\) is a superideal \(I\) of \(R\) such that \(I\not=R\). Let \(\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper superideals of \(R\). A homogeneous element \(a\in R\) is \(\phi\)-{\it prime} to \(I\) if \(ra\in I-\phi(I)\) where \(r\) is a homogeneous element in \(R\), then \(r\in I\). We denote by \(\nu_\phi(I)\) the set of all homogeneous elements in \(R\) that are not \(\phi\)-prime to \(I\). We define \(I\) to be \(\phi\)-\textit{primal} if the set \[P=\begin{cases}[(\nu_\phi(I))_0+(\nu_\phi(I))_1\cup\{0\}]+\phi(I) & :\quad {\rm if}\ \phi\not=\phi_\emptyset\\ (\nu_\phi(I))_0+(\nu_\phi(I))_1& :\quad {\rm if}\ \phi=\phi_\emptyset\end{cases}\]forms a superideal of \(R\). For example if we take \(\phi_\emptyset(I)=\emptyset\) (resp. \(\phi_0(I)=0\)), a \(\phi\)-\textit{primal} superideal is a primal superideal (resp., a weakly primal superideal). In this paper we study several generalizations of primal superideals of \(R\) and their properties. |
| first_indexed | 2025-12-02T15:26:28Z |
| format | Article |
| id | admjournalluguniveduua-article-26 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:26:28Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-262016-07-12T10:09:40Z Generalization of primal superideals Jaber, Ameer primal superideal, \(\phi\)-\(P\)-primal superideal, \(\phi\)-prime superideal 13A02, 16D25, 16W50 Let \(R\) be a commutative super-ring with unity \(1\not=0\). A proper superideal of \(R\) is a superideal \(I\) of \(R\) such that \(I\not=R\). Let \(\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper superideals of \(R\). A homogeneous element \(a\in R\) is \(\phi\)-{\it prime} to \(I\) if \(ra\in I-\phi(I)\) where \(r\) is a homogeneous element in \(R\), then \(r\in I\). We denote by \(\nu_\phi(I)\) the set of all homogeneous elements in \(R\) that are not \(\phi\)-prime to \(I\). We define \(I\) to be \(\phi\)-\textit{primal} if the set \[P=\begin{cases}[(\nu_\phi(I))_0+(\nu_\phi(I))_1\cup\{0\}]+\phi(I) & :\quad {\rm if}\ \phi\not=\phi_\emptyset\\ (\nu_\phi(I))_0+(\nu_\phi(I))_1& :\quad {\rm if}\ \phi=\phi_\emptyset\end{cases}\]forms a superideal of \(R\). For example if we take \(\phi_\emptyset(I)=\emptyset\) (resp. \(\phi_0(I)=0\)), a \(\phi\)-\textit{primal} superideal is a primal superideal (resp., a weakly primal superideal). In this paper we study several generalizations of primal superideals of \(R\) and their properties. Lugansk National Taras Shevchenko University 2016-07-12 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/26 Algebra and Discrete Mathematics; Vol 21, No 2 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/26/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/26/100 Copyright (c) 2016 Algebra and Discrete Mathematics |
| spellingShingle | primal superideal \(\phi\)-\(P\)-primal superideal \(\phi\)-prime superideal 13A02 16D25 16W50 Jaber, Ameer Generalization of primal superideals |
| title | Generalization of primal superideals |
| title_full | Generalization of primal superideals |
| title_fullStr | Generalization of primal superideals |
| title_full_unstemmed | Generalization of primal superideals |
| title_short | Generalization of primal superideals |
| title_sort | generalization of primal superideals |
| topic | primal superideal \(\phi\)-\(P\)-primal superideal \(\phi\)-prime superideal 13A02 16D25 16W50 |
| topic_facet | primal superideal \(\phi\)-\(P\)-primal superideal \(\phi\)-prime superideal 13A02 16D25 16W50 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/26 |
| work_keys_str_mv | AT jaberameer generalizationofprimalsuperideals |