The Zero Product Probability of Some Finite Ring of Matrices Based on the Order of the Annihilator

An annihilator is defined as the set of pairs of elements in a ring R in which the product of the elements in the pair is the zero element of R. In this paper, we aim to determine the order of the annihilator of the finite ring of matrices of dimension two over integers modulo prime, M2(Zp). The order of the annihilator is determined by using number theory, specifically the linear congruence method. Another subject that we discuss in this paper is the zero product probability of a finite ring. The zero product probability is defined as the probability that two elements of a finite ring have product zero. Based on the order of the annihilator, the general formula of the zero product probability of M2(Zp) is determined.