On nil clean group rings

Abstract

A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. In Sahinkaya et al. [13], it was shown that, for a ring R and a symmetric group S₃, the group ring RS₃ is nil clean iff R and $M_2(R)$ are nil clean. Let $D_{2n}$ be the dihedral group of order 2n and $Q_{2n}$ be the generalized quaternion group of order 2n. In this paper, we investigate a more general question and completely characterize when group rings $R{D}_{2n}$ and $R{Q}_{2n}$ are nil clean. It is proved that $R{D}_{2n}$ is nil clean iff, either $n=2^k$ and R is nil clean, or $n=3·2^k$ and RS₃ is nil clean, and a similar result is obtained for $R{Q}_{2n}.$ Furthermore, nil clean group rings with involution * are also investigated.

Keywords:

• Dihedral group
• generalized quaternion group
• group ring
• Nil clean ring
• nil *-clean ring

2010 Mathematics Subject Classification:

• 16S34
• 16U99
• 20D15
• 16W10

Acknowledgments

The authors would like to thank the referee for carefully reading the paper and for his/her valuable comments and suggestions, which help improving the readability of the paper.

Additional information

Funding

The research was carried out during a visit by the first author to Brock University as an international visiting scholar. He would like to gratefully thank the host institution for its hospitality and for providing an excellent atmosphere for research. This research was supported in part by Anhui Provincial Natural Science Foundation (Grant No. 2008085MA06) and the Key project of Anhui Education Committee (Grant No. gxyqZD2019009), and by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN 2017-03903).