Joshua Mac Intyre, a Master of Science candidate in the Department of Mathematics and Statistics, will virtually present the Masters Research Project titled Nil Clean Group Rings over Metacyclic Groups on Thursday, November 27, 2025 at 11:00 AM.
The examination committee includes Supervisor Dr. Yuanlin Li and Supervisory Committee Member Dr. Henryk Fukś.
Students (both graduate and undergraduate) as well as other members of the Brock Community are invited to attend. A Microsoft Teams link to the meeting can be found here: Join the meeting.
Keywords: Idempotent; group rings; nilpotent; nil clean; Peirce decomposition; Wedderburn-Artin; Metacyclic groups; Fermat numbers
Abstract: A ring is called nil clean if each element can be expressed as the sum of an idempotent and nilpotent. This presentation expounds on our work published in Nil clean group rings over metacyclic groups. We will assume at least an undergraduate understanding of group and ring theory but will provide some preliminary information on nil clean rings, group rings, and metacyclic groups. We will justify our comparison between the nil cleanness of ℤ2G, and any group ring RG, where R is a commutative ring and G is a finite group. This comparison will lead to an analysis of ℤ2G, for G up to an order of 20. Then, when considering a nil clean group ring RG over a metacyclic group G, we were able to reduce to the case where G = < a, b | a^{n} = b^{m} = 1, b^{-1}ab = a^{r} >, with m = 2k, n odd, and the center being trivial. We try to break it down a little further to when n is a prime power, calculating n for each m up to 16, and we verify whether most of these candidate group rings are indeed nil clean. Finally, we discuss the results of our investigation and a connection to Fermat numbers.
