On Idempotent Units in Commutative Group Rings

Authors : Ömer KÜSMÜŞ

Pages : 782-790

Doi:10.16984/saufenbilder.733935

View : 15 | Download : 8

Publication Date : 2020-08-01

Article Type : Research Paper

Abstract :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 idinsert ignore into journalissuearticles values(RG);={∑_insert ignore into journalissuearticles values(r_g∈idinsert ignore into journalissuearticles values(R););▒〖r_g g〗: ∑_insert ignore into journalissuearticles values(r_g∈idinsert ignore into journalissuearticles values(R););▒r_g =1 and r_g r_h=0 when g≠h} where idinsert ignore into journalissuearticles values(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.Vinsert ignore into journalissuearticles values(Rinsert ignore into journalissuearticles values(G×H););=idinsert ignore into journalissuearticles values(Rinsert ignore into journalissuearticles values(G×H););, ii.Vinsert ignore into journalissuearticles values(Rinsert ignore into journalissuearticles values(G×H););=G×idinsert ignore into journalissuearticles values(RH);, iii.Vinsert ignore into journalissuearticles values(Rinsert ignore into journalissuearticles values(G×H););=idinsert ignore into journalissuearticles values(RG);×H where 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].
Keywords : idempotent, unit, group ring, commutative