The Squared-Zero Product Probability of Some Ring of Matrices over Integers Modulo Prime

Recently, the study of probabilities in ring theory has shown a significant increase in the field of algebra. Many interesting algebraic structures were modeled to find their probabilities in certain finite rings. In this paper, we introduce a new type of probability in finite rings, namely the squared-zero product probability. The aim is to study the square property and the zero product property of the ring. The focus of this study is the ring of matrices of dimension two over integers modulo prime p. To obtain the probability, the order of the square-annihilator of the ring is determined. The results found show that the squared-zero product probability of the ring is dependent on the value of p.