On Idempotent Units in Commutative Group Rings
Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elem...
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Sakarya University
2020-08-01
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| Series: | Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/1210033 |
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| author | Ömer Küsmüş |
| author_facet | Ömer Küsmüş |
| author_sort | Ömer Küsmüş |
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| description | Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG)={∑_(r_g∈id(R))▒〖r_g g〗: ∑_(r_g∈id(R))▒r_g =1 and r_g r_h=0 when g≠h} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities:i.V(R(G×H))=id(R(G×H)),ii.V(R(G×H))=G×id(RH),iii.V(R(G×H))=id(RG)×Hwhere G×H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13]. |
| format | Article |
| id | doaj-art-034f1acb60bc4981a9db9148940e9aad |
| institution | Kabale University |
| issn | 2147-835X |
| language | English |
| publishDate | 2020-08-01 |
| publisher | Sakarya University |
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| series | Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi |
| spelling | doaj-art-034f1acb60bc4981a9db9148940e9aad2024-12-23T08:06:10ZengSakarya UniversitySakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi2147-835X2020-08-0124478279010.16984/saufenbilder.73393528On Idempotent Units in Commutative Group RingsÖmer Küsmüş0https://orcid.org/0000-0001-7397-0735VAN YÜZÜNCÜ YIL ÜNİVERSİTESİ, FEN FAKÜLTESİSpecial elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG)={∑_(r_g∈id(R))▒〖r_g g〗: ∑_(r_g∈id(R))▒r_g =1 and r_g r_h=0 when g≠h} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities:i.V(R(G×H))=id(R(G×H)),ii.V(R(G×H))=G×id(RH),iii.V(R(G×H))=id(RG)×Hwhere G×H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13].https://dergipark.org.tr/tr/download/article-file/1210033idempotentunitgroup ringcommutative |
| spellingShingle | Ömer Küsmüş On Idempotent Units in Commutative Group Rings Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi idempotent unit group ring commutative |
| title | On Idempotent Units in Commutative Group Rings |
| title_full | On Idempotent Units in Commutative Group Rings |
| title_fullStr | On Idempotent Units in Commutative Group Rings |
| title_full_unstemmed | On Idempotent Units in Commutative Group Rings |
| title_short | On Idempotent Units in Commutative Group Rings |
| title_sort | on idempotent units in commutative group rings |
| topic | idempotent unit group ring commutative |
| url | https://dergipark.org.tr/tr/download/article-file/1210033 |
| work_keys_str_mv | AT omerkusmus onidempotentunitsincommutativegrouprings |