On generalized probability in finite commutative rings

Authors : Shafiq Ur REHMAN, Muhammad Naveed SHAHERYAR

Pages : 125-132

Doi:10.24330/ieja.1156662

View : 14 | Download : 8

Publication Date : 2023-01-09

Article Type : Research Paper

Abstract :Let $R$ be a finite commutative ring with unity and $x\\in R$. We study the probability that the product of two randomly chosen elements insert ignore into journalissuearticles values(with replacement); of $R$ equals $x$. We denote this probability by $Prob_x insert ignore into journalissuearticles values(R);$. We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for $Prob_x insert ignore into journalissuearticles values(R);$ when $R=\\mathbb{Z}_m\\times \\mathbb{Z}_n$.
Keywords : Probability, finite commutative ring, Euler phi function, simple graph, zero divisor graph, complete graph, local ring