Groups of order $p^4$ as additive groups of local near-rings
UDC 512.6 Near-rings can be considered as a generalization of associative rings. In general, a near-ring is a ring $(R,+,\cdot)$ in which the operation $``+"$ is not necessarily Abelian and at least one distributive law holds. A near-ring with identity is called lo...
Gespeichert in:
| Datum: | 2024 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2024
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/8053 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: | |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512894005805056 |
|---|---|
| author | Raievska, I. Raievska, M. Раєвська, Ірина Раєвська, Марина |
| author_facet | Raievska, I. Raievska, M. Раєвська, Ірина Раєвська, Марина |
| author_sort | Raievska, I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-07-15T03:05:07Z |
| description | UDC 512.6
Near-rings can be considered as a generalization of associative rings. In general, a near-ring is a ring $(R,+,\cdot)$ in which the operation $``+"$ is not necessarily Abelian and at least one distributive law holds. A near-ring with identity is called local if the set of all invertible elements forms a subgroup of the additive group. In particular, every group is an additive group of some near-ring but not of a near-ring with identity. Finding non-Abelian finite $p$-groups that are additive groups of local near-rings is an open problem.
We considered groups of nilpotency class $2$ and $3$ of order $p^4$ as additive groups of local near-rings in\linebreak {\sf\scriptsize [https://arxiv.org/abs/2303.17567} and {\sf\scriptsize https://arxiv.org/abs/2309.14342]}. It was shown that, for $p>3,$ there exist local near-rings on one of  four nonisomorphic groups of nilpotency class $3$ of order $p^4$. In the present paper, we continue our investigation of the groups of nilpotency class $3$ of order $p^4$. In particular, it is shown that another group of this class is an additive group of a local near-ring and, hence, of a near-ring with identity.  Examples of local near-rings on this group have been constructed in the GAP computer algebra system. |
| doi_str_mv | 10.3842/umzh.v76i5.8053 |
| first_indexed | 2026-03-24T03:36:02Z |
| format | Article |
| fulltext |
Skip to main content
Skip to main navigation menu
Skip to site footer
Open Menu
Ukrains’kyi Matematychnyi Zhurnal
Current
Archives
Submissions
Major topics of interest
About
About Journal
Editorial Team
Ethics & Disclosures
Contacts
Search
Register
Login
Home
/
Login
Login
Required fields are marked with an asterisk: *
Subscription required to access item. To verify subscription, log in to journal.
Login
Username or Email
*
Required
Password
*
Required
Forgot your password?
Keep me logged in
Login
Register
Language
English
Українська
Information
For Readers
For Authors
For Librarians
subscribe
Subscribe
Latest publications
Make a Submission
Make a Submission
STM88 menghadirkan Link Gacor dengan RTP tinggi untuk peluang menang yang lebih sering! Bergabunglah sekarang dan buktikan keberuntungan Anda!
Pilih STM88 sebagai agen toto terpercaya Anda dan nikmati kenyamanan bermain dengan sistem betting cepat, result resmi, dan bonus cashback harian.
|
| id | umjimathkievua-article-8053 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:36:02Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/13/f7857ec01c4666b258959f37c96f6c13 |
| spelling | umjimathkievua-article-80532024-07-15T03:05:07Z Groups of order $p^4$ as additive groups of local near-rings Групи порядку $p^4$ як адитивні групи локальних майже-кілець Raievska, I. Raievska, M. Раєвська, Ірина Раєвська, Марина Local nearring Nearring with identity non-abelian group Локальне майже-кільце майже-кільце з одиницею неабелева група UDC 512.6 Near-rings can be considered as a generalization of associative rings. In general, a near-ring is a ring $(R,+,\cdot)$ in which the operation $``+"$ is not necessarily Abelian and at least one distributive law holds. A near-ring with identity is called local if the set of all invertible elements forms a subgroup of the additive group. In particular, every group is an additive group of some near-ring but not of a near-ring with identity. Finding non-Abelian finite $p$-groups that are additive groups of local near-rings is an open problem. We considered groups of nilpotency class $2$ and $3$ of order $p^4$ as additive groups of local near-rings in\linebreak {\sf\scriptsize [https://arxiv.org/abs/2303.17567} and {\sf\scriptsize https://arxiv.org/abs/2309.14342]}. It was shown that, for $p>3,$ there exist local near-rings on one of  four nonisomorphic groups of nilpotency class $3$ of order $p^4$. In the present paper, we continue our investigation of the groups of nilpotency class $3$ of order $p^4$. In particular, it is shown that another group of this class is an additive group of a local near-ring and, hence, of a near-ring with identity.  Examples of local near-rings on this group have been constructed in the GAP computer algebra system. УДК 512.6 Майже-кільця можна розглядати як узагальнення асоціативних кілець. У загальних рисах, майже-кільце — це кільце $(R,+,\cdot),$ де операція додавання необов'язково абелева та принаймні один дистрибутивний закон має місце. Майже-кільце з одиницею називається локальним, якщо множина всіх необоротних елементів утворює підгрупу в адитивній групі. Зокрема, кожна група є адитивною групою деякого майже-кільця, але не майже-кільця з одиницею. Визначення неабелевих скінченних $p$-груп, які є адитивними групами локальних майже-кілець, є відкритою проблемою. Групи класу нільпотентності $2$ та $3$ порядку $p^4$ як адитивні групи локальних майже-кілець розглядалися  в [{\sf\scriptsize https://arxiv.org/abs/2303.17567} та {\sf\scriptsize https://arxiv.org/abs/2309.14342}]. Було показано, що для $p>3$ існують локальні майже-кільця на одній з чотирьох неізоморфних груп класу нільпотентності $3$ порядку $p^4.$ У цій статті продовжено дослідження груп класу нільпотентності $3$ порядку $p^4.$ Зокрема, показано, що ще одна з цих груп є адитивною групою локального майже-кільця, а отже майже-кільця з одиницею. В системі комп'ютерної алгебри GAP побудовано приклади локальних майже-кілець на цій групі.  Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8053 10.3842/umzh.v76i5.8053 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 890–906 Український математичний журнал; Том 76 № 6 (2024); 890–906 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/8053/10035 Copyright (c) 2024 Maryna Raievska |
| spellingShingle | Raievska, I. Raievska, M. Раєвська, Ірина Раєвська, Марина Groups of order $p^4$ as additive groups of local near-rings |
| title | Groups of order $p^4$ as additive groups of local near-rings |
| title_alt | Групи порядку $p^4$ як адитивні групи локальних майже-кілець |
| title_full | Groups of order $p^4$ as additive groups of local near-rings |
| title_fullStr | Groups of order $p^4$ as additive groups of local near-rings |
| title_full_unstemmed | Groups of order $p^4$ as additive groups of local near-rings |
| title_short | Groups of order $p^4$ as additive groups of local near-rings |
| title_sort | groups of order $p^4$ as additive groups of local near-rings |
| topic_facet | Local nearring Nearring with identity non-abelian group Локальне майже-кільце майже-кільце з одиницею неабелева група |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8053 |
| work_keys_str_mv | AT raievskai groupsoforderp4asadditivegroupsoflocalnearrings AT raievskam groupsoforderp4asadditivegroupsoflocalnearrings AT raêvsʹkaírina groupsoforderp4asadditivegroupsoflocalnearrings AT raêvsʹkamarina groupsoforderp4asadditivegroupsoflocalnearrings AT raievskai grupiporâdkup4âkaditivnígrupilokalʹnihmajžekílecʹ AT raievskam grupiporâdkup4âkaditivnígrupilokalʹnihmajžekílecʹ AT raêvsʹkaírina grupiporâdkup4âkaditivnígrupilokalʹnihmajžekílecʹ AT raêvsʹkamarina grupiporâdkup4âkaditivnígrupilokalʹnihmajžekílecʹ |