On Symmetric Elements and Symmetric Units in Group Rings
2006; Taylor & Francis; Volume: 34; Issue: 2 Linguagem: Inglês
10.1080/00927870500388018
ISSN1532-4125
AutoresEric Jespers, Manuel Ruiz Marín,
Tópico(s)Rings, Modules, and Algebras
ResumoABSTRACT Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi (2001 Bovdi , V. ( 2001 ). On symmetric units in group algebras . Comm. Algebra 29 ( 12 ): 5411 – 5422 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Bovdi et al. (1996 Bovdi , V. , Kovács , L. G. , Sehgal , S. K. (1996). Symmetric units in modular group algebras. Comm. Algebra 24(3):803–808. [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Bovdi and Parmenter (1997 Bovdi , V. , Parmenter , M. M. ( 1997 ). Symmetric units in integral group rings . Publ. Math. Debrecen 50 ( 3–4 ): 369 – 372 . [CSA] [Google Scholar]), Broche Cristo (2003 Broche Cristo , O. ( 2003 ). A Commutatividade dos Elementos Simétricos e Anti-simétricos en Anéis de Grupo, Ph.D. thesis, Univ. Sao Paulo . [Google Scholar], to appear), Giambruno and Sehgal (1993 Giambruno , A. , Sehgal , S. K. ( 1993 ). Lie nilpotence of group rings . Comm. Algebra 21 : 4253 – 4261 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Lee (1999 Lee , G. T. ( 1999 ). Group rings whose symmetric elements are Lie nilpotent . Proc. Amer. Math. Soc. 127 ( 11 ): 3153 – 3159 . [CROSSREF] [CSA] [Crossref] , [Google Scholar]), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g−1.
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