On the sum of an idempotent and a tripotent in a quaternion algebra over the ring of integers modulo p

Department of Mathematical and Actuarial Science, Universiti Tunku Abdul Rahman, Jalan Sungai Long, Bandar Sungai Long, 43000 Kajang, Selangor

^* e-mail: shiro271828@1utar.my
^** e-mail: quakt@utar.edu.my
^*** e-mail: deniswong@utar.edu.my

Abstract

Let H be denoted as quaternions. Quaternions form an algebra over a ring R, as an extension of complex numbers into a four dimensional space, where H = {a₀ + a₁i + a₂j + a₃k | a₀, a₁, a₂, a₃ ∈ R}. A quaternion algebra, particularly defined over fields of characteristic 0, finds numerous applications in physics. In this article, we explore some properties of the sum of an idempotent and a tripotent in the finite ring H/Z_p, adapting the definition of SIT rings that was introduced by Ying et al in 2016. We provide some conditions for H/Z_p to be SIT rings and we give some examples of weakly tripotent rings (Breaz and Cimpean, 2018) in H/Z_p.