On semiperfect $a$-rings
UDC 512.5 A ring is  called a right $a$-ring if  every right ideal is automorphism invariant.  We describe some properties of $a$-rings over  semiperfect rings.   It is shown that an  I-finite right $a$-ring&a...
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| Date: | 2024 |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7491 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512675871588352 |
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| author | Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu |
| author_facet | Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu |
| author_sort | Van, Truong Thi Thuy |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-07-15T03:05:04Z |
| description | UDC 512.5
A ring is  called a right $a$-ring if  every right ideal is automorphism invariant.  We describe some properties of $a$-rings over  semiperfect rings.   It is shown that an  I-finite right $a$-ring  is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if $R$ is  an indecomposable (as a ring) I-finite right $a$-ring not  simple with nontrivial idempotents  such that  every minimal right ideal  is a right annihilator and  ${\rm Soc}(R_R)={\rm Soc}(_RR)$  is essential in $R_R$, then $R$ is a quasi-Frobenius ring and it is also  a right $q$-ring.  |
| doi_str_mv | 10.3842/umzh.v76i5.7491 |
| first_indexed | 2026-03-24T03:32:34Z |
| format | Article |
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| id | umjimathkievua-article-7491 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:34Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-74912024-07-15T03:05:04Z On semiperfect $a$-rings On semiperfect $a$-rings Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu semiperfect ring, Nakayama permutation, automorphism-invariant module UDC 512.5 A ring is  called a right $a$-ring if  every right ideal is automorphism invariant.  We describe some properties of $a$-rings over  semiperfect rings.   It is shown that an  I-finite right $a$-ring  is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if $R$ is  an indecomposable (as a ring) I-finite right $a$-ring not  simple with nontrivial idempotents  such that  every minimal right ideal  is a right annihilator and  ${\rm Soc}(R_R)={\rm Soc}(_RR)$  is essential in $R_R$, then $R$ is a quasi-Frobenius ring and it is also  a right $q$-ring.  УДК 512.5 Про напівдосконалі $a$-кільця Кільце називається правим $a$-кільцем, якщо кожний правий ідеал є інваріантним щодо автоморфізму.  Ми описуємо деякі властивості $a$-кілець над напівдосконалими кільцями. Показано, що I-скінченне праве $a$-кільце є прямою сумою напівпростого артинового кільця та базисного кільця. Показано також, що якщо $R$ є нерозкладним (як кільце) I-скінченним правим $a$-кільцем, яке не є простим з нетривіальними ідемпотентами, тобто таким, що кожен мінімальний правий ідеал є правим анігілятором, а ${\rm Soc}(R_R)={\rm Soc}(_RR)$ є істотним в $R_R$, то $R$ є не лише квазіфробеніусовим кільцем, а й правим $q$-кільцем.  Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7491 10.3842/umzh.v76i5.7491 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 907–914 Український математичний журнал; Том 76 № 6 (2024); 907–914 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7491/10036 Copyright (c) 2024 Thi Thuy Van Truong |
| spellingShingle | Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu Van, Truong Thi Thuy Alghamdi, Ahmad M. Alkinani, Amnah Abdu On semiperfect $a$-rings |
| title | On semiperfect $a$-rings |
| title_alt | On semiperfect $a$-rings |
| title_full | On semiperfect $a$-rings |
| title_fullStr | On semiperfect $a$-rings |
| title_full_unstemmed | On semiperfect $a$-rings |
| title_short | On semiperfect $a$-rings |
| title_sort | on semiperfect $a$-rings |
| topic_facet | semiperfect ring Nakayama permutation automorphism-invariant module |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7491 |
| work_keys_str_mv | AT vantruongthithuy onsemiperfectarings AT alghamdiahmadm onsemiperfectarings AT alkinaniamnahabdu onsemiperfectarings AT vantruongthithuy onsemiperfectarings AT alghamdiahmadm onsemiperfectarings AT alkinaniamnahabdu onsemiperfectarings |