On center graphs of finite associative rings
UDC 512.5 We consider a finite associative ring $R,$ which may have or may not have a unit element. We also examine its center denoted by $Z(R).$  Our main focus is on the introduction of two distinct graphs associated with $R,$ namely, the center graph denoted...
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| Datum: | 2024 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7391 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512665162481664 |
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| author | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. |
| author_facet | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. |
| author_sort | Jorf, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-07-15T03:05:00Z |
| description | UDC 512.5
We consider a finite associative ring $R,$ which may have or may not have a unit element. We also examine its center denoted by $Z(R).$  Our main focus is on the introduction of two distinct graphs associated with $R,$ namely, the center graph denoted by $GC(R)$ and the strict center graph denoted by $\overline{GC(R)}.$
We present the properties of $GC(R)$ and explore its implications on the nature of $Z(R).$  Specifically, we demonstrate that if $GC(R)$ is complete, then $Z(R)$ is an ideal in $R.$  Furthermore, in the case where $R$ is a unital ring, the completeness of $GC(R)$ leads to the conclusion that $R$ is a commutative ring.
As a specific application of our results, we provide an explicit construction of the graph $\overline{GC}(T_2(p)),$ where $T_2(p)$ represents the ring of upper-triangular matrices with entries in the ring $\mathbb{Z}/p\mathbb{Z}$ and $p$ is a prime integer.
In our investigations of the center graph and strict center graph, we aim to shed light on the properties of finite associative rings and their centers, thus providing valuable insights and applications in the ring theory. |
| doi_str_mv | 10.3842/umzh.v76i5.7391 |
| first_indexed | 2026-03-24T03:32:24Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7391 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:24Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-73912024-07-15T03:05:00Z On center graphs of finite associative rings On center graphs of finite associative rings Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. ideal center commutativity UDC 512.5 We consider a finite associative ring $R,$ which may have or may not have a unit element. We also examine its center denoted by $Z(R).$  Our main focus is on the introduction of two distinct graphs associated with $R,$ namely, the center graph denoted by $GC(R)$ and the strict center graph denoted by $\overline{GC(R)}.$ We present the properties of $GC(R)$ and explore its implications on the nature of $Z(R).$  Specifically, we demonstrate that if $GC(R)$ is complete, then $Z(R)$ is an ideal in $R.$  Furthermore, in the case where $R$ is a unital ring, the completeness of $GC(R)$ leads to the conclusion that $R$ is a commutative ring. As a specific application of our results, we provide an explicit construction of the graph $\overline{GC}(T_2(p)),$ where $T_2(p)$ represents the ring of upper-triangular matrices with entries in the ring $\mathbb{Z}/p\mathbb{Z}$ and $p$ is a prime integer. In our investigations of the center graph and strict center graph, we aim to shed light on the properties of finite associative rings and their centers, thus providing valuable insights and applications in the ring theory. УДК 512.5 Про центральні графи скінченних асоціативних кілець  Розглянуто скінченне асоціативне кільце $R,$ яке може мати або не мати одиничний елемент, та досліджено його центр, позначений як $Z(R).$ Основну увагу приділено введенню двох різних графів, пов'язаних з $R,$ а саме центрального графа, позначеного як $GC(R),$ і  строго центрального графа, позначеного як $\overline{GC(R)}.$ Наведено властивості $GC(R)$ і досліджено їхні наслідки для природи $Z(R).$ Зокрема, показано, що у випадку, коли $GC(R)$ є повним,  $Z(R)$ є ідеалом в $R.$ Навіть більше, у випадку, коли $R$ є унітарним кільцем, повнота $GC(R)$ приводить до висновку, що $R$ є комутативним кільцем. Як конкретне застосування одержаних результатів наведено явну конструкцію графа $\overline{GC}(T_2(p)),$ де $T_2(p)$ — кільце верхньотрикутних матриць з елементами, що належать кільцю $\mathbb{Z}/p\mathbb{Z},$ а $p$ --- просте ціле число. Досліджуючи центральний граф і строгий центральний граф, ми прагнемо пролити світло на властивості скінченних асоціативних кілець та їхніх центрів і отримати цінні висновки і застосування в теорії кілець. Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7391 10.3842/umzh.v76i5.7391 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 843–854 Український математичний журнал; Том 76 № 6 (2024); 843–854 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7391/10031 Copyright (c) 2024 mohamed jorf |
| spellingShingle | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. On center graphs of finite associative rings |
| title | On center graphs of finite associative rings |
| title_alt | On center graphs of finite associative rings |
| title_full | On center graphs of finite associative rings |
| title_fullStr | On center graphs of finite associative rings |
| title_full_unstemmed | On center graphs of finite associative rings |
| title_short | On center graphs of finite associative rings |
| title_sort | on center graphs of finite associative rings |
| topic_facet | ideal center commutativity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7391 |
| work_keys_str_mv | AT jorfm oncentergraphsoffiniteassociativerings AT oukhtitel oncentergraphsoffiniteassociativerings AT jorfm oncentergraphsoffiniteassociativerings AT oukhtitel oncentergraphsoffiniteassociativerings |