On Symmetric Elements and Symmetric Units in Group Rings

ABSTRACT

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 et al. (1996), Bovdi and Parmenter (1997), Broche Cristo (2003, to appear), Giambruno and Sehgal (1993), and Lee (1999), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g⁻¹.

Key Words:

• Group ring
• Involution
• Symmetric elements

2000 AMS Subject Classification:

• 16U60
• 16W10
• 16S34
• 20C07

ACKNOWLEDGMENTS

The authors would like to thank the referee for some comments to improve the article.

The author Eric Jespers has been partially supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Vlaanderen), Flemish-Polish bilateral agreement BIL 01/31.

The author Manuel Ruiz Maŕin has been partially supported by PFMPDI-UPCT-2003 and Fundación Séneca.

Notes

Communicated by M. Ferrero.