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

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/Zp, adapting the definition of SIT rings that was introduced by Ying et al in 2016. We provide some conditions for H/Zp to be SIT rings and we give some examples of weakly tripotent rings (Breaz and Cimpean, 2018) in H/Zp.