Open AccessTHE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY
Author(s) -
Nurhidayah Zaid,
Nor Haniza Sarmin,
Sanhan Muhammad Salih Khasraw
Publication year - 2020
Publication title -
jurnal teknologi/jurnal teknologi
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.191
H-Index - 22
eISSN - 2180-3722
pISSN - 0127-9696
DOI - 10.11113/jurnalteknologi.v83.14936
Subject(s) - zero divisor , mathematics , combinatorics , commutative ring , distance regular graph , zero (linguistics) , discrete mathematics , graph , regular graph , voltage graph , modulo , line graph , commutative property , linguistics , philosophy
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.