Trivial Extension of π-Regular Rings
In this paper we investigate if it is possible that the trivial extension ring T(R,R) inherit the properties of the ring R and present the relationship between the trivial extension T(R,M) of a ring R by an R-module M and theπ-regularity of Rby taking new concepts asπ-coherent rings and C-π-regular...
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| Format: | Article |
| Language: | English |
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Unviversity of Technology- Iraq
2016-01-01
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| Series: | Engineering and Technology Journal |
| Subjects: | |
| Online Access: | https://etj.uotechnology.edu.iq/article_112594_6d98539910be9b18fd260e4754445423.pdf |
| _version_ | 1797325957285019648 |
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| author | Areej M. Abduldaim |
| author_facet | Areej M. Abduldaim |
| author_sort | Areej M. Abduldaim |
| collection | DOAJ |
| description | In this paper we investigate if it is possible that the trivial extension ring T(R,R) inherit the properties of the ring R and present the relationship between the trivial extension T(R,M) of a ring R by an R-module M and theπ-regularity of Rby taking new concepts asπ-coherent rings and C-π-regular rings whichintroducedas extensions of the concept of π-regularrings. Moreover we studied the possibility of being the trivial extensionT(R,M) itselfπ-regular ring according to specific conditions. Thus we proved that if R is an Artinian ring, then the trivial extension T(R,R) is a π-regular ring. As well as ifthe trivial extension T(R,R) is a Noetherian π-regularring, then R is a π-regular ring. On the other hand we showed that if F is a field and M is an F-vector space with infinite dimension, then the trivial extension ring T(F,M) of F by M is C-π-regular ring. |
| first_indexed | 2024-03-08T06:16:44Z |
| format | Article |
| id | doaj.art-ad4e5b31aa4e448b8463cbc3ef9f12fc |
| institution | Directory Open Access Journal |
| issn | 1681-6900 2412-0758 |
| language | English |
| last_indexed | 2024-03-08T06:16:44Z |
| publishDate | 2016-01-01 |
| publisher | Unviversity of Technology- Iraq |
| record_format | Article |
| series | Engineering and Technology Journal |
| spelling | doaj.art-ad4e5b31aa4e448b8463cbc3ef9f12fc2024-02-04T17:27:18ZengUnviversity of Technology- IraqEngineering and Technology Journal1681-69002412-07582016-01-01341B15315910.30684/etj.34.1B.17112594Trivial Extension of π-Regular RingsAreej M. AbduldaimIn this paper we investigate if it is possible that the trivial extension ring T(R,R) inherit the properties of the ring R and present the relationship between the trivial extension T(R,M) of a ring R by an R-module M and theπ-regularity of Rby taking new concepts asπ-coherent rings and C-π-regular rings whichintroducedas extensions of the concept of π-regularrings. Moreover we studied the possibility of being the trivial extensionT(R,M) itselfπ-regular ring according to specific conditions. Thus we proved that if R is an Artinian ring, then the trivial extension T(R,R) is a π-regular ring. As well as ifthe trivial extension T(R,R) is a Noetherian π-regularring, then R is a π-regular ring. On the other hand we showed that if F is a field and M is an F-vector space with infinite dimension, then the trivial extension ring T(F,M) of F by M is C-π-regular ring.https://etj.uotechnology.edu.iq/article_112594_6d98539910be9b18fd260e4754445423.pdftrivial extension ringregular ringartinian ringnoetherian ring |
| spellingShingle | Areej M. Abduldaim Trivial Extension of π-Regular Rings Engineering and Technology Journal trivial extension ring regular ring artinian ring noetherian ring |
| title | Trivial Extension of π-Regular Rings |
| title_full | Trivial Extension of π-Regular Rings |
| title_fullStr | Trivial Extension of π-Regular Rings |
| title_full_unstemmed | Trivial Extension of π-Regular Rings |
| title_short | Trivial Extension of π-Regular Rings |
| title_sort | trivial extension of π regular rings |
| topic | trivial extension ring regular ring artinian ring noetherian ring |
| url | https://etj.uotechnology.edu.iq/article_112594_6d98539910be9b18fd260e4754445423.pdf |
| work_keys_str_mv | AT areejmabduldaim trivialextensionofpregularrings |