Generalized symmetric rings
In this paper, we introduce a class of rings which is a generalization of symmetric rings. Let \(R\) be a ring with identity. A ring \(R\) is called central symmetric if for any \(a\), \(b, c\in R\), \(abc = 0\) implies bac belongs to the center of \(R\). Since every symmetric ring is central sy...
Gespeichert in:
| Datum: | 2018 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
|
| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/683 |
| 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-683 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua:article-6832018-04-04T09:31:27Z Generalized symmetric rings Kafkas, G. Ungor, B. Halicioglu, S. Harmanci, A. symmetric rings, central reduced rings, central symmetric rings, central reversible rings, central semicommutative rings, central Armendariz rings, 2-primal rings 13C99, 16D80, 16U80 In this paper, we introduce a class of rings which is a generalization of symmetric rings. Let \(R\) be a ring with identity. A ring \(R\) is called central symmetric if for any \(a\), \(b, c\in R\), \(abc = 0\) implies bac belongs to the center of \(R\). Since every symmetric ring is central symmetric, we study sufficient conditions for central symmetric rings to be symmetric. We prove that some results of symmetric rings can be extended to central symmetric rings for this general settings. We show that every central reduced ring is central symmetric, every central symmetric ring is central reversible, central semmicommutative, 2-primal, abelian and so directly finite. It is proven that the polynomial ring \(R[x]\) is central symmetric if and only if the Laurent polynomial ring \(R[x, x^{-1}]\) is central symmetric. Among others, it is shown that for a right principally projective ring \(R\), \(R\) is central symmetric if and only if \(R[x]/(x^n)\) is central Armendariz, where \(n\geq 2~\) is a natural number and \((x^n)\) is the ideal generated by \(x^n\). Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/683 Algebra and Discrete Mathematics; Vol 12, No 2 (2011) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/683/217 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2018-04-04T09:31:27Z |
| collection |
OJS |
| language |
English |
| topic |
symmetric rings central reduced rings central symmetric rings central reversible rings central semicommutative rings central Armendariz rings 2-primal rings 13C99 16D80 16U80 |
| spellingShingle |
symmetric rings central reduced rings central symmetric rings central reversible rings central semicommutative rings central Armendariz rings 2-primal rings 13C99 16D80 16U80 Kafkas, G. Ungor, B. Halicioglu, S. Harmanci, A. Generalized symmetric rings |
| topic_facet |
symmetric rings central reduced rings central symmetric rings central reversible rings central semicommutative rings central Armendariz rings 2-primal rings 13C99 16D80 16U80 |
| format |
Article |
| author |
Kafkas, G. Ungor, B. Halicioglu, S. Harmanci, A. |
| author_facet |
Kafkas, G. Ungor, B. Halicioglu, S. Harmanci, A. |
| author_sort |
Kafkas, G. |
| title |
Generalized symmetric rings |
| title_short |
Generalized symmetric rings |
| title_full |
Generalized symmetric rings |
| title_fullStr |
Generalized symmetric rings |
| title_full_unstemmed |
Generalized symmetric rings |
| title_sort |
generalized symmetric rings |
| description |
In this paper, we introduce a class of rings which is a generalization of symmetric rings. Let \(R\) be a ring with identity. A ring \(R\) is called central symmetric if for any \(a\), \(b, c\in R\), \(abc = 0\) implies bac belongs to the center of \(R\). Since every symmetric ring is central symmetric, we study sufficient conditions for central symmetric rings to be symmetric. We prove that some results of symmetric rings can be extended to central symmetric rings for this general settings. We show that every central reduced ring is central symmetric, every central symmetric ring is central reversible, central semmicommutative, 2-primal, abelian and so directly finite. It is proven that the polynomial ring \(R[x]\) is central symmetric if and only if the Laurent polynomial ring \(R[x, x^{-1}]\) is central symmetric. Among others, it is shown that for a right principally projective ring \(R\), \(R\) is central symmetric if and only if \(R[x]/(x^n)\) is central Armendariz, where \(n\geq 2~\) is a natural number and \((x^n)\) is the ideal generated by \(x^n\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/683 |
| work_keys_str_mv |
AT kafkasg generalizedsymmetricrings AT ungorb generalizedsymmetricrings AT halicioglus generalizedsymmetricrings AT harmancia generalizedsymmetricrings |
| first_indexed |
2025-07-17T10:33:50Z |
| last_indexed |
2025-07-17T10:33:50Z |
| _version_ |
1837889938787401728 |