[This article belongs to Volume - 38, Issue - 07]

The Zero Product Probability of Ring of Matrices Based on Euler’s Phi-Function

In the field of algebra, the application of probability theory in ring theory has been widely studied by various researchers. In this paper, a type of probability in finite rings, namely the zero product probability is determined for some ring of matrices with a single nonzero entry. To obtain the zero product probability, the exact order of the annihilator of a ring R needs to be first determined. The annihilator of R is defined as the set of pairs of elements, where the product of elements in each pair is the zero element of R. The general formula for the exact order is established using the linear congruence method as well as Euler’s phi-function. The zero product probability of R is then found by dividing the exact order of the annihilator by the square of the order of R. Besides that, a subset of the annihilator, which is the square-annihilator that only focuses on the square attributes of the annihilator is also determined for the same ring. The exact order of the square-annihilator is then used to find the squared-zero product probability of R.