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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| Format: | Article |
| Language: | English |
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Sakarya University
2020-08-01
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| Series: | Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/tr/download/article-file/1210033 |
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| Summary: | 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]. |
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| ISSN: | 2147-835X |