# THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY

## DOI:

https://doi.org/10.11113/jurnalteknologi.v83.14936## Keywords:

Zero divisor, ring theory, ring of matrices, graph theory, zero divisor graph## Abstract

Let *R* be a finite ring. The zero divisors of *R* are defined as two nonzero elements of *R*, say *x* and *y* where *xy *=* *0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo *n*. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of *R* and two distinct vertices *x* and *y* are adjacent if and only if *xy* = 0. It is found that the zero divisor graph of *R* is a directed graph.

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*Jurnal Teknologi*,

*83*(1), 127–132. https://doi.org/10.11113/jurnalteknologi.v83.14936

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